The stability analysis of a co-circulation model for COVID-19, dengue, and zika with nonlinear incidence rates and vaccination strategies

Abstract

This paper aims to study the impacts of COVID-19 and dengue vaccinations on the dynamics of zika transmission by developing a vaccination model with the incorporation of saturated incidence rates. Analyses are performed to assess the qualitative behavior of the model. Carrying out bifurcation analysis of the model, it was concluded that co-infection, super-infection and also re-infection with same or different disease could trigger backward bifurcation. Employing well-formulated Lyapunov functions, the model’s equilibria are shown to be globally stable for a certain scenario. Moreover, global sensitivity analyses are performed out to assess the impact of dominant parameters that drive each disease’s dynamics and its co-infection. Model fitting is performed on the actual data for the state of Amazonas in Brazil. The fittings reveal that our model behaves very well with the data. The significance of saturated incidence rates on the dynamics of three diseases is also highlighted. Based on the numerical investigation of the model, it was observed that increased vaccination efforts against COVID-19 and dengue could positively impact zika dynamics and the co-spread of triple infections.

1. Introduction

Dengue virus (DENV) is the most common mosquito-borne flavivirus which has affected more than half of the world’s population in more than 125 countries [1]. Dengue viruses are antigenically categorized into four serotypes that are phylogenetically descended from a common ancestor and cause common pathologies in humans. Infection with any serotype of dengue virus may be symptomatic (25%) or not [2]. Most symptomatic cases develop only limited febrile illness with high fever, myalgia, arthralgia, headache, facial flushing, rash, orbital pain, vomiting, epistaxis, or gingival bleeding. The febrile phase lasts between 2 to 7 days, whereas viremia and plasma NS1 can be detected in the first 1–4 days. Only a small proportion of symptomatic infected persons develop complications with dengue hemorrhagic fever (DHF), and dengue shock syndrome (DSS) which result in leakage of the plasma and bleeding. On the other hand, zika virus (ZIKV) has recently emerged as a great public health concern. It began with its initial isolation from a monkey in the forest of zika in Uganda in 1947 [3]. Subsequently, only a few outbreaks were recorded in some parts of Asia and Africa, with the highest number of cases not more than few dozens, during each outbreak, until 2007 [4], [5]. Symptoms of zika disease include: fever, arthralgia, rash, and conjunctivitis but more than 50% of cases were without symptoms [6]. However, the threat posed by zika was not given much attention until around 2015, when it caused a severe outbreaks in Brazil [7]. During this time, more than 35% of the reported cases in Brazil were from the northeastern Brazilian region [8], where zika virus infected more than 60% of Salvadorans.

DENV and ZIKV are both transmitted by Aedes mosquitoes, mainly A. aegypti and A. albopictus [9]. These are responsible to produce diseases with some similar but diverse clinical features. Recent experimental evidences show the cross-reactivity between zika and dengue [10], [11]. Indeed, the pre-dengue plasma was able to influence antibody-dependent enhancement of ZIKV. Co-circulation of multiple DENV serotypes and the co-endemicity of DENV and ZIKV have been reported [12]. In particular, co-dynamics of DENV and ZIKV were noticed in some patients in New Caledonia in 2014, during the ZIKV outbreak [13]. Co-circulations could pose potential threat to public health given the fact that more than one-third of the world’s population is domiciled in countries where DENV is endemic [14]. Therefore, it is a matter of great concern for public health agencies to ascertain how the available vaccination program against COVID-19 and DENV affects ZIKV transmission when the co-endemicity of all diseases becomes widespread.

On the other hand, the Coronavirus disease 2019 (COVID-19) is a respiratory illness caused by severe acute respiratory syndrome coronavirus 2 (SARSCoV-2) [15] and has become endemic now. In tropical and subtropical areas of the world, where arboviruses and COVID-19 can coexist as a result of geographic intersection of both diseases, laboratory diagnosis has become difficult and patients need screening for both viruses. Owing to intersecting symptoms of both COVID-19 and arboviruses, there is always the tendency for wrong diagnosis [16]. Co-infections between COVID-19 and DENV/ZIKV have been detected in many countries [17]. Patients with DENV or ZIKV, simultaneously infected with COVID-19, could suffer severe illness and hospitalization [17]. It is worth noting that individuals co-infected with COVID-19 and DENV may have very high glucose levels which could lead to increased COVID-19 spread [18]. Moreover, high mortality is always related with patients co-infected with both COVID-19 and arboviruses [18]. The COVID-19 pandemic has greatly misled the diagnosis, treatment and vaccination campaigns, putting millions of lives at risk of vaccine-preventable diseases [19]. Also, the COVID-19 pandemic resulted in reduced access to humanitarian assistance, increased the pressure on already weak health systems in DENV/ZIKV-endemic regions, and has made difficult to handle two or more concurrent outbreaks [19].

Since the advent of COVID-19 pandemic, different vaccines have been approved against the disease, including BNT162b2 (Pfizer/ BioNTech), mRNA-1273 (Moderna), AZD-1222/ChAdOx1-nCoV (Oxford/AstraZeneca), Ad26.COV2.S (Janssen by Johnson and Johnson), rAd26-S+rAd5-S (Sputnik V), NVX-CoV2373 (Novavax), CoronaVac (Sinovac) and many others [20]. The effectiveness of the approved vaccines ranges from 60% to more than 90% [20]. In 2015, a new DENV vaccine, called Dengvaxia, was developed (which has been approved for usage in several countries) [21], [22], [23], [24]. Effectiveness of the tetravalent vaccine varies from one serotype to the other. For instance, it has 54.7% effectiveness for serotype 1, 43.0% effectiveness for serotype 2, 71.6% efficacy for serotype 3 and 76.9% efficacy for serotype 4 [22], [25], [26]. Though the precise effect of the DENV vaccine on ZIKV has not been accurately estimated, yet some studies have pointed out the effectiveness of the DENV vaccine against ZIKV [11], [13].

Mathematical models have become instrumental in exploring the behavior of infectious diseases [27], [28], [29], [30], [31], [32], [33], [34], [35], [36]. Many models have investigated the interactions between COVID-19 and other infectious diseases such as hepatitis B virus [37], influenza [38], malaria [39], [40], dengue [41], zika [42], diabetes [43], tuberculosis [44], HIV [45]. Recently, authors in [46] have analyzed a model for COVID-19, chikungunya, zika and dengue with re-infection which has neither considered vaccination nor super-infection between the arboviruses and hence motivates further study to fill the gaps. Also, it is well known that the number of effective contacts between infectious and susceptible humans may saturate at high infective levels due to crowding effect of infectious individuals or due to the precautionary measures put in place by the uninfected individuals [47], [48]. The incidence rate to describe this is termed the saturated (or Holling-type II) incidence and has been successfully applied in many disease models [47], [49], [50]. In mathematical models dealing with high co-endemicity of COVID-19 and arboviruses, this may serve as the best form of incidence rate. Moreover, super-infection has been used to describe the interaction between pathogen communities sharing a common host [51], [52].

To fill up the gaps in the study of disease co-dynamics, we have designed a new vaccination model for the co-circulation of COVID-19, DENV and ZIKV incorporating saturated incidence rates and aim to contribute in the following ways:

• (i.)Analyze qualitatively for the occurrence of backward bifurcation.
• (ii.)Employ well constructed Lyapunov functions to investigate the stability of equilibria.
• (iii.)Determine the parameters which drive the dynamics of the diseases and their co-infections through global sensitivity analysis of the model.
• (iv.)Study the suitability of the proposed model with reference to real COVID-19, DENV and ZIKV data.
• (v.)Discuss the impact of saturated incidence rates and various control measures in curtailing the triple infections.

2. Model formulation

We first define some notations needed in the sequel. At any time $t$, the total population of humans ${\mathcal{N}}_h\left(t\right)$ composed of these states: unvaccinated susceptible humans ${\mathcal{S}}_h\left(t\right)$, humans vaccinated against COVID-19 ${\mathcal{S}}_{hc}\left(t\right)$, humans vaccinated against DENV, but with some vaccine-derived immunity against ZIKV infection ${\mathcal{S}}_{hd}\left(t\right)$, humans infected with COVID-19 ${\mathcal{I}}_c\left(t\right)$, humans infected with DENV ${\mathcal{I}}_d\left(t\right)$, humans infected with ZIKV ${\mathcal{I}}_z\left(t\right)$, co-infected humans with both COVID-19 and DENV ${\mathcal{I}}_{cd}\left(t\right)$, co-infected humans with both COVID-19 and ZIKV ${\mathcal{I}}_{cz}\left(t\right)$, and humans who have recovered from COVID-19, DENV, ZIKV or co-infections $\mathcal{R}\left(t\right)$. The total vector population, at any time $t$, ${\mathcal{N}}_v\left(t\right)$ consists vectors that are susceptible: ${\mathcal{S}}_v\left(t\right)$, vectors infected with DENV, ${\mathcal{I}}_d^v\left(t\right)$ and vectors infected with ZIKV, ${\mathcal{I}}_z^v\left(t\right)$. Recruitment into human classes is given by ${\Lambda }_h$. Susceptible humans, ${\mathcal{S}}_h$ can get infected with COVID-19, DENV or ZIKV infection at the rates, $\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}$, $\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}$ and $\frac{{\beta }_3{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z}$, respectively. Human natural death rate is given by ${\mu }_h$. Co-infection death rates are given by ${\varphi }_c,{\varphi }_d$ and ${\varphi }_z$, respectively. Individuals in COVID-19 infected compartment can infected with DENV or ZIKV at the rates ${\delta }_1\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}$ and ${\delta }_2\frac{{\beta }_3{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z}$, respectively. Similarly, those in DENV and ZIKV compartments can get additional infection with COVID-19 at the rates ${\delta }_3\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}$ and ${\delta }_4\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}$, respectively. Due to super-infection between diseases caused by Aedes aegypti [51], we have assumed compartments with the co-infection of only two diseases (one of which must be COVID-19). It is assumed in this model that DENV infection can increase ZIKV dynamics through antibody-dependent enhancement. Also, adopting the assumption in [51], we have assumed that the vaccine against DENV can have some efficacy against incident infection with ZIKV.

Recovery rates for COVID-19, DENV and ZIKV infected individuals is given by ${\zeta }_c$, ${\zeta }_d$, and ${\zeta }_z$ respectively. We have also assumed recovery from concurrent infections with COVID-19 and DENV or COVID-19 and ZIKV at the rates ${\zeta }_{cd}$ and ${\zeta }_{cz}$, respectively. For the sake of convenience, and to avoid complexity of the model, we have assumed that rate of getting re-infection for recovered and the rate of incident infection for susceptible individuals are the same. Recovery from one disease does not give an individual protection against infection with another disease. However, this present model has some limitations. To avoid complexity, asymptomatic classes of COVID-19, dengue and zika were not taken into consideration. Furthermore, individual recovery classes were not considered to avoid major model complications. They can be included in further research. The model’s flow chart and description of parameters in the model are given in Fig. 1 and Table 1, respectively.

$$\frac{d{\mathcal{S}}_h\left(t\right)}{dt}={\Lambda }_h-\left(\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}+\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}+\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}\right)\left(+{\mu }_h+{\theta }_1+{\theta }_2\right){\mathcal{S}}_h,$$

$$\frac{d{\mathcal{S}}_{hc}\left(t\right)}{dt}={\theta }_1{\mathcal{S}}_h-\left((1-{\eta }_1)\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}+\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}\right)\left(+\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}+{\mu }_h\right){\mathcal{S}}_{hc},$$

$$\frac{d{\mathcal{S}}_{hd}\left(t\right)}{dt}={\theta }_2{\mathcal{S}}_h-\left(\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}+(1-{\eta }_2)\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}\right)$$

$$\left(+(1-{\eta }_3)\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}+{\mu }_h\right){\mathcal{S}}_{hd},$$

$$\frac{d{\mathcal{I}}_c\left(t\right)}{dt}=\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}({\mathcal{S}}_h+(1-{\eta }_1){\mathcal{S}}_{hc}+{\mathcal{S}}_{hd}+\mathcal{R})-\left({\varphi }_c+{\zeta }_c+{\mu }_h\right){\mathcal{I}}_c-{\delta }_1\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}{\mathcal{I}}_c-{\delta }_2\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}{\mathcal{I}}_c,$$

$$\frac{d{\mathcal{I}}_d\left(t\right)}{dt}=\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_2){\mathcal{S}}_{hd}+\mathcal{R})+\xi \frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}{\mathcal{I}}_z-\left({\varphi }_d+{\zeta }_d+{\mu }_h\right){\mathcal{I}}_d-{\delta }_3\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}{\mathcal{I}}_d,$$

$$\frac{d{\mathcal{I}}_z\left(t\right)}{dt}=\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_3){\mathcal{S}}_{hd}+\mathcal{R})-\left({\varphi }_z+{\zeta }_z+{\mu }_h\right){\mathcal{I}}_z-{\delta }_4\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}{\mathcal{I}}_z-\xi \frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}{\mathcal{I}}_z,$$

$$\frac{d{\mathcal{I}}_{cd}\left(t\right)}{dt}={\delta }_1\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}{\mathcal{I}}_c+{\delta }_3\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}{\mathcal{I}}_d-\left({\varphi }_{cd}+{\zeta }_{cd}+{\mu }_h\right){\mathcal{I}}_{cd},$$

$$\frac{d{\mathcal{I}}_{cz}\left(t\right)}{dt}={\delta }_2\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}{\mathcal{I}}_c+{\delta }_4\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}{\mathcal{I}}_z-\left({\varphi }_{cz}+{\zeta }_c+{\mu }_h\right){\mathcal{I}}_{cz},$$

$$\frac{d\mathcal{R}\left(t\right)}{dt}={\zeta }_c{\mathcal{I}}_c+{\zeta }_d{\mathcal{I}}_d+{\zeta }_z{\mathcal{I}}_z+{\zeta }_{cd}{\mathcal{I}}_{cd}+{\zeta }_{cz}{\mathcal{I}}_{cz}-\left(\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}}+\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}\right)\left(+\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}+{\mu }_h\right)\mathcal{R},$$$$\frac{d{\mathcal{S}}_v\left(t\right)}{dt}={\Lambda }_v-\left[\frac{{\beta }_2^v({\mathcal{I}}_d+{\mathcal{I}}_{cd})}{1+{\vartheta }_2{\mathcal{I}}_d+{\vartheta }_{12}{\mathcal{I}}_{cd}}+\frac{{\beta }_3^v({\mathcal{I}}_z+{\mathcal{I}}_{cz})}{1+{\vartheta }_3{\mathcal{I}}_z+{\vartheta }_{13}{\mathcal{I}}_{cz}}+{\mu }_v\right]{\mathcal{S}}_v,$$$$\frac{d{\mathcal{I}}_d^v\left(t\right)}{dt}=\frac{{\beta }_2^v({\mathcal{I}}_d+{\mathcal{I}}_{cd})}{1+{\vartheta }_2{\mathcal{I}}_d+{\vartheta }_{12}{\mathcal{I}}_{cd}}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_d^v,$$$$\frac{d{\mathcal{I}}_z^v\left(t\right)}{dt}=\frac{{\beta }_3^v({\mathcal{I}}_z+{\mathcal{I}}_{cz})}{1+{\vartheta }_3{\mathcal{I}}_z+{\vartheta }_{13}{\mathcal{I}}_{cz}}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_z^v,$$ (1)

subject to the initial conditions

$${\mathcal{S}}_{h0}={\mathcal{S}}_h\left(0\right),{\mathcal{S}}_{hc0}={\mathcal{S}}_{hc}\left(0\right),{\mathcal{S}}_{hd0}={\mathcal{S}}_{hd}\left(0\right),{\mathcal{I}}_{c0}={\mathcal{I}}_c\left(0\right),$$

$${\mathcal{I}}_{d0}={\mathcal{I}}_d\left(0\right),{\mathcal{I}}_{z0}={\mathcal{I}}_z\left(0\right){\mathcal{I}}_{cd0}={\mathcal{I}}_{cd}\left(0\right),$$

$${\mathcal{I}}_{cz0}={\mathcal{I}}_{cz}\left(0\right),{\mathcal{R}}_0=\mathcal{R}\left(0\right),{\mathcal{S}}_{v0}={\mathcal{S}}_v\left(0\right),{\mathcal{I}}_{d0}^v={\mathcal{I}}_d^v\left(0\right),{\mathcal{I}}_{z0}^v={\mathcal{I}}_z^v\left(0\right).$$

Fig. 1.

Flow charts of the model (1), with ${\lambda }_1=(1-{\eta }_1)\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}},{\lambda }_2=(1-{\eta }_2)\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}$.

Table 1.

Description of parameters in the model (1).

Parameter	Description	Value	Source
${\Lambda }_h$	Recruitment rate for humans	$\frac{4,269,995}{74.9\times 365}$	[53]
${\Lambda }_v$	Recruitment rate for vectors	1,500	Assumed
${\mathrm{\mu }}_h$	Human natural death rate	$\frac{1}{74.9\times 365}$$da{y}^{-1}$	[53]
${\mathrm{\mu }}_v$	Vector removal rate	$\frac{1}{21}$$da{y}^{-1}$	[46]
${\beta }_1$	COVID-19 transmission rate	$8.5065\times 10^{-8}$$da{y}^{-1}$	Fitted
${\beta }_2^h$	Infection rate for vector-human
	transmission of DENV	$2.5675\times 10^{-8}$$da{y}^{-1}$	Fitted
${\beta }_2^v$	Infection rate for human-vector
	transmission of DENV	$5.0\times 10^{-6}$$da{y}^{-1}$	Fitted
${\beta }_3^h$	Infection rate for vector-human
	transmission of ZIKV	$5.2218\times 10^{-10}$$da{y}^{-1}$	Fitted
${\beta }_3^v$	Infection rate for human-vector
	transmission of ZIKV	$5.3\times 10^{-6}$$da{y}^{-1}$	Fitted
${\zeta }_c$	COVID-19 recovery rate	$[\frac{1}{30},\frac{1}{3}]$	[54]
${\zeta }_d$	DENV recovery rate	$0.09-0.15$	[51]
${\zeta }_z$	ZIKV recovery rate	$0.09-0.15$	[51]
${\zeta }_{cd},{\zeta }_{cz}$	Co-infection recovery rate	$0.09-0.15$	Assumed
${\varphi }_c$	COVID-19-induced death rate	0.015 /day	[55]
${\varphi }_d$	DENV induced death rate	0.005	[51]
${\varphi }_z$	ZIKV induced death rates	0.005	[51]
${\varphi }_{cd}$	co-infection death rates	0.005	Assumed
${\varphi }_{cz}$	co-infection death rates	0.015	[42]
${\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4$	rate of co-infection with a second disease	1.0	Assumed
${\vartheta }_1,{\vartheta }_2,{\vartheta }_3$	1.0	Assumed
$\xi$	Super-infection parameter	(0.07-0.41) $da{y}^{-1}$	[51]
${\theta }_1$	COVID-19 vaccination rate	0.15 $da{y}^{-1}$	Assumed
${\theta }_2$	Dengue vaccination rate	(0.15-0.48) $da{y}^{-1}$	[51]
${\eta }_1$	COVID-19 vaccine efficacy	0.85–0.95	[56]
${\eta }_2$	DENV vaccine efficacy	0.81–0.88	[51]
${\eta }_3$	DENV vaccine efficacy against ZIKV	0.74–0.88	[51]
${\mathcal{R}}_{0c}$	COVID-19 associated reproduction number	1.2773	Estimated
${\mathcal{R}}_{0d}$	DENV associated reproduction number	1.5920	Estimated
${\mathcal{R}}_{0z}$	ZIKV associated reproduction number	0.2374	Estimated

3. Analysis of the model

In this section, the qualitative analysis of the model is performed.

3.1. Non-negativity of the model solutions

To show the model (1) is epidemiologically meaningful, it is appropriate to prove that the model solutions are non-negative over the time. Let us start with the following:

Theorem 3.1

Given the initial states ${\mathcal{S}}_h\left(0\right)\ge 0,{\mathcal{S}}_{hc}\left(0\right)\ge 0,{\mathcal{S}}_{hd}\left(0\right)\ge 0,{\mathcal{I}}_c\left(0\right)\ge 0,{\mathcal{I}}_d\left(0\right)\ge 0,{\mathcal{I}}_z\left(0\right)\ge 0,{\mathcal{I}}_{cd}\left(0\right)\ge 0,{\mathcal{I}}_{cz}\left(0\right)\ge 0,\mathcal{R}\left(0\right)\ge 0,{\mathcal{S}}_v\left(0\right)\ge 0,{\mathcal{I}}_d^v\left(0\right)\ge 0,{\mathcal{I}}_z^v\left(0\right)\ge 0$ .

Then the solutions, $({\mathcal{S}}_h\left(t\right),{\mathcal{S}}_{hc}\left(t\right),{\mathcal{S}}_{hd}\left(t\right),{\mathcal{I}}_c\left(t\right),{\mathcal{I}}_d\left(t\right),{\mathcal{I}}_z\left(t\right),{\mathcal{I}}_{cd}\left(t\right),{\mathcal{I}}_{cz}\left(t\right),\mathcal{R}\left(t\right),{\mathcal{S}}_v\left(t\right),{\mathcal{I}}_d^v\left(t\right),{\mathcal{I}}_z^v\left(t\right))$ of the model (1) are non-negative for all time $t>0$ .

Proof

Let

$t_f=sup\{t>0:{\mathcal{S}}_h\left(0\right)>0,{\mathcal{S}}_{hc}\left(0\right)>0,{\mathcal{S}}_{hd}\left(0\right)>0,{\mathcal{I}}_c\left(0\right)>0,{\mathcal{I}}_d\left(0\right)>0,{\mathcal{I}}_z\left(0\right)>0,{\mathcal{I}}_{cd}\left(0\right)>0,{\mathcal{I}}_{cz}\left(0\right)>0,\mathcal{R}\left(0\right)>0,{\mathcal{S}}_v\left(0\right)>0,{\mathcal{I}}_d^v\left(0\right)>0,{\mathcal{I}}_z^v\left(0\right)>0\}$.

From the 1st equation of system (1), we have

$$\frac{d{\mathcal{S}}_h\left(t\right)}{dt}={\Lambda }_h-({\lambda }_c\left(t\right)+{\lambda }_d^v\left(t\right)+{\lambda }_z^v\left(t\right)+{\mu }_h+{\theta }_1+{\theta }_2){\mathcal{S}}_h$$ (2)

where,

$${\lambda }_c\left(t\right)=\frac{{\beta }_1({\mathcal{I}}_c+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz})}{1+{\vartheta }_1{\mathcal{I}}_c+{\vartheta }_{12}{\mathcal{I}}_{cd}+{\vartheta }_{13}{\mathcal{I}}_{cz}},{\lambda }_d^v\left(t\right)=\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v},{\lambda }_z^v=\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}.$$

Applying the integrating factor method on (2), we obtain that

$$\frac{d}{dt}\left\{{\mathcal{S}}_h\left(t\right)exp\left[{\int }_0^t({\lambda }_c\left(\sigma \right)+{\lambda }_d^v\left(\sigma \right)+{\lambda }_z^v\left(\sigma \right))d\sigma +({\mu }_h+{\theta }_1+{\theta }_2)t\right]\right\}={\Lambda }_{h}exp\left[{\int }_0^t({\lambda }_c\left(\sigma \right)+{\lambda }_d^v\left(\sigma \right)+{\lambda }_z^v\left(\sigma \right))d\sigma +({\mu }_h+{\theta }_1+{\theta }_2)t\right]$$ (3)

Integrating both sides of (3) gives that

$${\mathcal{S}}_h\left(t_f\right)exp\left[{\int }_0^{t_f}({\lambda }_c\left(\sigma \right)+{\lambda }_d^v\left(\sigma \right)+{\lambda }_z^v\left(\sigma \right))d\sigma +({\mu }_h+{\theta }_1+{\theta }_2)t_f\right]-S\left(0\right)={\Lambda }_h{\int }_0^{t_f}exp\left[{\int }_0^x({\lambda }_c\left(\sigma \right)+{\lambda }_d^v\left(\sigma \right)+{\lambda }_z^v\left(\sigma \right))d\sigma +({\mu }_h+{\theta }_1+{\theta }_2)x\right]dx.$$

Thus,

$${\mathcal{S}}_h\left(t_f\right)={\mathcal{S}}_h\left(0\right)exp\left[-{\int }_0^{t_f}({\lambda }_c\left(\sigma \right)+{\lambda }_d^v\left(\sigma \right)+{\lambda }_z^v\left(\sigma \right))d\sigma -({\mu }_h+{\theta }_1+{\theta }_2)t_f\right]+exp\left[-{\int }_0^{t_f}({\lambda }_c\left(\sigma \right)+{\lambda }_d^v\left(\sigma \right)+{\lambda }_z^v\left(\sigma \right))d\sigma -({\mu }_h+{\theta }_1+{\theta }_2)t_f\right]\times {\Lambda }_h{\int }_0^{t_f}exp\left[{\int }_0^x({\lambda }_c\left(\sigma \right)+{\lambda }_d^v\left(\sigma \right)+{\lambda }_z^v\left(\sigma \right))d\sigma +({\mu }_h+{\theta }_1+{\theta }_2)x\right]dx$$

$$\ge 0.$$

Hence, ${\mathcal{S}}_h\left(t\right)\ge 0$ for all time $t>0$.

Similarly, it can be shown that:

${\mathcal{S}}_{hc}\left(t\right)\ge 0,{\mathcal{S}}_{hd}\left(t\right)\ge 0,{\mathcal{I}}_c\left(t\right)\ge 0,{\mathcal{I}}_d\left(t\right)\ge 0,{\mathcal{I}}_z\left(t\right)\ge 0,{\mathcal{I}}_{cd}\left(t\right)\ge 0,{\mathcal{I}}_{cz}\left(t\right)\ge 0,\mathcal{R}\left(t\right)\ge 0,{\mathcal{S}}_v\left(t\right)\ge 0,{\mathcal{I}}_d^v\left(t\right)\ge 0,{\mathcal{I}}_z^v\left(t\right)\ge 0$.

3.2. Boundedness of the solution

Theorem 3.2

The closed set $\mathcal{D}={\mathcal{D}}_h\times {\mathcal{D}}_v$ with

$${\mathcal{D}}_h=.\{({\mathcal{S}}_h,{\mathcal{S}}_{hc},{\mathcal{S}}_{hd},{\mathcal{I}}_c,{\mathcal{I}}_d,{\mathcal{I}}_z,{\mathcal{I}}_{cd},{\mathcal{I}}_{cz},\mathcal{R})\in {\mathfrak{R}}_{+}^9:$$

$${\mathcal{S}}_h+{\mathcal{S}}_{hc}+{\mathcal{S}}_{hd}+{\mathcal{I}}_c+{\mathcal{I}}_d+{\mathcal{I}}_z+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz}+\mathcal{R}\le \frac{{\Lambda }_h}{{\mu }_h}.\},$$

$${\mathcal{D}}_v=.\left\{({\mathcal{S}}_v,{\mathcal{I}}_d^v,{\mathcal{I}}_z^v)\in {\mathfrak{R}}_{+}^3:{\mathcal{S}}_v+{\mathcal{I}}_d^v+{\mathcal{I}}_z^v\le \frac{{\Lambda }_v}{{\mu }_v}.\right\}.$$

is positively invariant with respect to the model (1) .

Proof

If all equations relating to human components of system (1) are added up, it gives

$$\frac{d{\mathcal{N}}_h\left(t\right)}{dt}={\Lambda }_h-{\mu }_h{\mathcal{N}}_h\left(t\right)-[{\varphi }_c{\mathcal{I}}_c+{\varphi }_d{\mathcal{I}}_d+{\varphi }_z{\mathcal{I}}_z+{\varphi }_{cd}{\mathcal{I}}_{cd}+{\varphi }_{cz}{\mathcal{I}}_{cz}].$$ (4)

From (4), we have

$$\frac{d{\mathcal{N}}_h\left(t\right)}{dt}\le {\Lambda }_h-{\mu }_h{\mathcal{N}}_h\left(t\right),$$

that is,

$$\frac{d{\mathcal{N}}_h\left(t\right)}{dt}+\mu {\mathcal{N}}_h\left(t\right)\le \Lambda ,$$ (5)

By applying the integrating factor method to (5) and simplifying, we obtain the following inequality:

$${\mathcal{N}}_h\left(t\right)\le \frac{\Lambda }{\mu }+\left({\mathcal{N}}_h\left(0\right)-\frac{\Lambda }{\mu }\right)e^{-\mu t},$$

which further implies that

$$lim\underset{t\to \infty }{sup}{\mathcal{N}}_h\left(t\right)\le \frac{\Lambda }{\mu }.$$ (6)

Therefore, ${\mathcal{N}}_h\left(t\right)\le \frac{{\Lambda }_h}{{\mu }_h}$ as $t\to \infty$. Following similar arguments as above, it can be shown that ${\mathcal{N}}_v\left(t\right)\left(t\right)\le \frac{{\Lambda }_v}{{\mu }_v}$. Hence, the system (1) has the solution in $\mathcal{D}$ and hence the given system is positively invariant.

3.3. The basic reproduction number of the model

By setting the right-hand sides of the equations in the model (1) equal to zero, the disease free equilibrium (DFE) of the model (1) is obtained thus

$${\mathcal{G}}_0=\left({\mathcal{S}}_h^{\ast },{\mathcal{S}}_{hc}^{\ast },{\mathcal{S}}_{hd}^{\ast },{\mathcal{I}}_c^{\ast },{\mathcal{I}}_d^{\ast },{\mathcal{I}}_z^{\ast },{\mathcal{I}}_{cd}^{\ast },{\mathcal{I}}_{cz}^{\ast },{\mathcal{R}}^{\ast },{\mathcal{S}}_v^{\ast },{\mathcal{I}}_d^{v\ast },{\mathcal{I}}_z^{v\ast }\right)=\left(\frac{{\Lambda }_h}{({\mu }_h+{\theta }_1+{\theta }_2)},\frac{{\theta }_1{\Lambda }_h}{{\mu }_h({\mu }_h+{\theta }_1+{\theta }_2)},\frac{{\theta }_2{\Lambda }_h}{{\mu }_h({\mu }_h+{\theta }_1+{\theta }_2)},0,0,0,0,0,0,\right)$$

$$\left(\frac{{\Lambda }_v}{{\mu }_v},0,0\right)$$

The model’s reproduction number is calculated by adopting the next generation operator principle [57] on system (1). The transfer matrices are given by

$$F=\left(\begin{array}{ccccccc}\hfill {\beta }_1{\mathcal{A}}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_1{\mathcal{A}}^{\ast }\hfill & \hfill {\beta }_1{\mathcal{A}}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_2^h{\mathcal{B}}^{\ast }\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_3^h{\mathcal{C}}^{\ast }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\beta }_2^v{\Lambda }_v}{{\mu }_v}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{{\beta }_3^v{\Lambda }_v}{{\mu }_v}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill \end{array}\right),V=\left(\begin{array}{ccccccc}\hfill {\mathcal{K}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathcal{K}}_2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{K}}_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{K}}_4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{K}}_5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\mu }_v\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\mu }_v\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill \end{array}\right),$$ (7)

where,

$${\mathcal{A}}^{\ast }={\mathcal{S}}_h^{\ast }+(1-{\eta }_1){\mathcal{S}}_{hc}^{\ast }+{\mathcal{S}}_{hd}^{\ast }=\frac{{\Lambda }_h[{\mu }_h+(1-{\eta }_1){\theta }_1+{\theta }_2]}{{\mu }_h({\mu }_h+{\theta }_1+{\theta }_2)},$$

$${\mathcal{B}}^{\ast }={\mathcal{S}}_h^{\ast }+{\mathcal{S}}_{hc}^{\ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast }=\frac{{\Lambda }_h[{\mu }_h+{\theta }_1+(1-{\eta }_2){\theta }_2]}{{\mu }_h({\mu }_h+{\theta }_1+{\theta }_2)},$$

$${\mathcal{C}}^{\ast }={\mathcal{S}}_h^{\ast }+{\mathcal{S}}_{hc}^{\ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast }=\frac{{\Lambda }_h[{\mu }_h+{\theta }_1+(1-{\eta }_3){\theta }_2]}{{\mu }_h({\mu }_h+{\theta }_1+{\theta }_2)},$$

$${\mathcal{K}}_1={\varphi }_c+{\zeta }_c+{\mu }_h,{\mathcal{K}}_2={\varphi }_d+{\zeta }_d+{\mu }_h,{\mathcal{K}}_3={\varphi }_z+{\zeta }_z+{\mu }_h,$$

$${\mathcal{K}}_4={\varphi }_{cd}+{\zeta }_{cd}+{\mu }_h,{\mathcal{K}}_5={\varphi }_{cz}+{\zeta }_{cz}+{\mu }_h.$$

The basic reproduction number of the model (1) is given by ${\mathcal{R}}_0=\rho \left(F{V}^{-1}\right)=\text{max}\{{\mathcal{R}}_{0c},{\mathcal{R}}_{0d},{\mathcal{R}}_{0z}\}$ where ${\mathcal{R}}_{0c}$, ${\mathcal{R}}_{0d}$ and ${\mathcal{R}}_{0z}$ are the reproduction numbers related to COVID-19, dengue and zika virus, and are given by

$${\mathcal{R}}_{0c}=\frac{{\beta }_1{\Lambda }_h[{\mu }_h+(1-{\eta }_1){\theta }_1+{\theta }_2]}{{\mu }_h({\mu }_h+{\theta }_1+{\theta }_2)({\varphi }_c+{\zeta }_c+{\mu }_h)},$$

$${\mathcal{R}}_{0d}=\sqrt{\frac{{\beta }_2^h{\beta }_2^v{\Lambda }_h{\Lambda }_v[{\mu }_h+{\theta }_1+(1-{\eta }_2){\theta }_2]}{{\mu }_h{\mu }_v^2({\mu }_h+{\theta }_1+{\theta }_2)({\varphi }_d+{\zeta }_d+{\mu }_h)}},$$

$${\mathcal{R}}_{0z}=\sqrt{\frac{{\beta }_3^h{\beta }_3^v{\Lambda }_h{\Lambda }_v[{\mu }_h+{\theta }_1+(1-{\eta }_3){\theta }_2]}{{\mu }_h{\mu }_v^2({\mu }_h+{\theta }_1+{\theta }_2)({\varphi }_z+{\zeta }_z+{\mu }_h)}}.$$

Epidemiologically, the basic reproduction numbers above are interpreted as the average number of secondary infections which are generated by a single infected individual (with any of the three diseases) in a completely susceptible population [57].

3.4. Local asymptotic stability of the disease free equilibrium (DFE) of the model

Theorem 3.3

The (DFE) ${\mathcal{G}}_0$ of the model (1) is locally asymptotically stable (LAS) if ${\mathcal{R}}_0<1$ and unstable if ${\mathcal{R}}_0>1$ .

Proof

The model’s stability in the neighborhood of the infection free-equilibrium is analyzed by the with the aid of the Jacobian matrix of the system given in (see Box I)

with $\mathcal{H}=({\mu }_h+{\theta }_1+{\theta }_2),{\beta }_{11}=(1-{\eta }_1){\beta }_1,{\beta }_{22}^h=(1-{\eta }_2){\beta }_2^h,{\beta }_{33}^h=(1-{\eta }_3){\beta }_3^h$.

The eigenvalues are given by:

$${\varrho }_1=-({\varphi }_{cd}+{\zeta }_{cd}+{\mu }_h),{\varrho }_2=-({\varphi }_{cz}+{\zeta }_{cz}+{\mu }_h),{\varrho }_3=-({\mu }_h+{\theta }_1+{\theta }_2),$$

$${\varrho }_4=-{\mu }_h\left(\text{with multiplicity of}3\right),$$

$${\varrho }_5=-{\mu }_v,$$

whereas, the rest are the solutions obtained from:

$$\varrho +{\mathcal{K}}_1(1-{\mathcal{R}}_{0c})=0,$$ (8)

$${\varrho }^2+({\mu }_v+{\mathcal{K}}_2)\varrho +{\mu }_v{\mathcal{K}}_2(1-{\mathcal{R}}_{0d}^2)=0,\text{ and}$$ (9)

$${\varrho }^2+({\mu }_v+{\mathcal{K}}_3)\varrho +{\mu }_v{\mathcal{K}}_3(1-{\mathcal{R}}_{0z}^2)=0,$$ (10)

Adopting the Routh–Hurwitz criterion, all three Eqs. (8)–(10) will have roots (not greater than or equal zero) if and only if the reproduction numbers ${\mathcal{R}}_{0c}<1$, ${\mathcal{R}}_{0d}<1$ and ${\mathcal{R}}_{0z}<1$. Thus, ${\mathcal{G}}_0$ is locally asymptotically stable if ${\mathcal{R}}_0=\text{max}\{{\mathcal{R}}_{0c},{\mathcal{R}}_{0d},{\mathcal{R}}_{0z}\}<1$.

In terms of Epidemiology, Theorem 3.3 can be interpreted as follows: the co-circulation of the triple infections can be eradicated from the population if the threshold, ${\mathcal{R}}_0<1$ and the initial sizes of the sub-populations of the model (1) are in the neighborhood of the DFE (${\mathcal{G}}_0$). In other words, the introduction of few infected persons into the population will not cause large outbreak of the diseases, and the spread will recede over the time.

Box I.

$$\left(\begin{array}{cccccccccccc}\hfill -\mathcal{H}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_1{\mathcal{S}}_h^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_1{\mathcal{S}}_h^{\ast }\hfill & \hfill -{\beta }_1{\mathcal{S}}_h^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_2^h{\mathcal{S}}_h^{\ast }\hfill & \hfill -{\beta }_3^h{\mathcal{S}}_h^{\ast }\hfill \\ \hfill {\theta }_1\hfill & \hfill -{\mu }_h\hfill & \hfill 0\hfill & \hfill -{\beta }_{11}{\mathcal{S}}_{hc}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_{11}{\mathcal{S}}_{hc}^{\ast }\hfill & \hfill -{\beta }_{11}{\mathcal{S}}_{hc}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_2^h{\mathcal{S}}_{hc}^{\ast }\hfill & \hfill -{\beta }_3^h{\mathcal{S}}_{hc}^{\ast }\hfill \\ \hfill {\theta }_2\hfill & \hfill 0\hfill & \hfill -{\mu }_h\hfill & \hfill -{\beta }_1{\mathcal{S}}_{hd}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_1{\mathcal{S}}_{hd}^{\ast }\hfill & \hfill -{\beta }_1{\mathcal{S}}_{hd}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_{22}^h{\mathcal{S}}_{hd}^{\ast }\hfill & \hfill -{\beta }_{33}^h{\mathcal{S}}_{hd}^{\ast }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_1{\mathcal{A}}^{\ast }-{\mathcal{K}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_1{\mathcal{A}}^{\ast }\hfill & \hfill {\beta }_1{\mathcal{A}}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mathcal{K}}_2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_2^h{\mathcal{B}}^{\ast }\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mathcal{K}}_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_3^h{\mathcal{C}}^{\ast }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mathcal{K}}_4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mathcal{K}}_5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\zeta }_c\hfill & \hfill {\zeta }_d\hfill & \hfill {\zeta }_z\hfill & \hfill {\zeta }_{cd}\hfill & \hfill {\zeta }_{cz}\hfill & \hfill -{\mu }_h\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_2^v{\mathcal{S}}_v^{\ast }\hfill & \hfill -{\beta }_3^v{\mathcal{S}}_v^{\ast }\hfill & \hfill -{\beta }_2^v{\mathcal{S}}_v^{\ast }\hfill & \hfill -{\beta }_3^v{\mathcal{S}}_v^{\ast }\hfill & \hfill 0\hfill & \hfill -{\mu }_v\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_2^v{\mathcal{S}}_v^{\ast }\hfill & \hfill 0\hfill & \hfill {\beta }_2^v{\mathcal{S}}_v^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mu }_v\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_3^v{\mathcal{S}}_v^{\ast }\hfill & \hfill 0\hfill & \hfill {\beta }_3^v{\mathcal{S}}_v^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mu }_v\hfill \end{array}\right),$$

3.5. Backward bifurcation analysis of the model

In this section, we perform the backward bifurcation analysis of the proposed model (1). The scenario of backward bifurcation, which has been explored in a lot of epidemic models, is usually indicated by the coexistence of a stable disease-free equilibrium and a stable endemic equilibrium when the associated model reproduction number is less than one. Epidemiologically, the occurrence of this phenomenon implies that the common epidemiological requirement that the reproduction number ${\mathcal{R}}_0$ to be less than one, while necessary, is no longer sufficient for effective disease control. The Center Manifold theory by Castillo-Chavez and Song [58] (provided in the Appendix) will be our guide in establishing whether backward bifurcation occurs in the model considered in this work or not.

We claim the result below:

Theorem 3.4

The model (1) will undergo backward bifurcation if the condition below holds:

$$a=2{\omega }_4{\nu }_4\text{{}{\beta }_1[{\omega }_1+(1-{\eta }_1){\omega }_2+{\omega }_3-2{\vartheta }_1{\mathcal{A}}^{\ast }{\omega }_4+{\omega }_9]-[{\delta }_1{\beta }_2^h{\omega }_{11}+{\delta }_2{\beta }_3^h{\omega }_{11}]\text{}}+2{\beta }_2^h{\omega }_{11}{\nu }_5\{{\omega }_1+{\omega }_2+(1-{\eta }_2){\omega }_3-2{\vartheta }_2{\mathcal{B}}^{\ast }{\omega }_{11}+\xi {\omega }_6+{\omega }_9\}-2{\beta }_1{\delta }_3{\omega }_4{\omega }_5{\nu }_5+2{\beta }_3^h{\omega }_{12}{\nu }_6\{{\omega }_1+{\omega }_2+(1-{\eta }_3){\omega }_3-2{\vartheta }_3{\mathcal{C}}^{\ast }{\omega }_{12}+{\omega }_9\}-2{\beta }_1{\delta }_4{\omega }_4{\omega }_6{\nu }_6-\xi {\beta }_2^h{\omega }_6{\omega }_{11}{\nu }_6-2{\beta }_2^v({\omega }_5+{\omega }_7){\nu }_{11}({\vartheta }_2{\omega }_5{x}_{10}^{\ast }+{\vartheta }_{12}{\omega }_7{x}_{10}^{\ast }-{\omega }_{10})-2{\beta }_3^v({\omega }_6+{\omega }_8){\nu }_{12}({\vartheta }_2{\omega }_6{x}_{10}^{\ast }+{\vartheta }_{13}{\omega }_8{x}_{10}^{\ast }-{\omega }_{10})+2{\omega }_4{\nu }_7({\delta }_1{\beta }_2^h{\omega }_{11}+{\delta }_3{\beta }_1{\omega }_5)+2{\omega }_4{\nu }_8({\delta }_2{\beta }_2^h{\omega }_{12}+{\delta }_4{\beta }_1{\omega }_6)>0.$$

Proof

Let

$${\mathcal{K}}_e=\left({\mathcal{S}}_h^{\ast \ast },{\mathcal{S}}_{hc}^{\ast \ast },{\mathcal{S}}_{hd}^{\ast \ast },{\mathcal{I}}_c^{\ast \ast },{\mathcal{I}}_d^{\ast \ast },{\mathcal{I}}_z^{\ast \ast },{\mathcal{I}}_{cd}^{\ast \ast },{\mathcal{I}}_{cz}^{\ast \ast },{\mathcal{R}}^{\ast \ast },{\mathcal{S}}_v^{\ast \ast },{\mathcal{I}}_d^{v\ast \ast },{\mathcal{I}}_z^{v\ast \ast }\right)$$

represent any endemic equilibrium of system (1).

Change the variables as follows

$${\mathcal{S}}_h=x_1,{\mathcal{S}}_{hc}=x_2,{\mathcal{S}}_{hd}=x_3,{\mathcal{I}}_c=x_4,{\mathcal{I}}_d=x_5,{\mathcal{I}}_z=x_6,{\mathcal{I}}_{cd}=x_7,{\mathcal{I}}_{cz}=x_8,\mathcal{R}=x_9,{\mathcal{S}}_v=x_{10},{\mathcal{I}}_d^v=x_{11},{\mathcal{I}}_z^v=x_{12},$$

the system (1) can be re-presented in the form below:

$$\frac{d{x}_1}{dt}={\Lambda }_h-\left(\frac{{\beta }_1(x_4+x_7+x_8)}{1+{\vartheta }_1{x}_4+{\vartheta }_{12}{x}_7+{\vartheta }_{13}{x}_8}+\frac{{\beta }_2^h{x}_{11}}{1+{\vartheta }_2{x}_{11}}+\frac{{\beta }_3^h{x}_{12}}{1+{\vartheta }_3{x}_{12}}\right)\left(+{\mu }_h+{\theta }_1+{\theta }_2\right)x_1\equiv f_1,$$

$$\frac{d{x}_2}{dt}={\theta }_1{x}_1-\left((1-{\eta }_1)\frac{{\beta }_1(x_4+x_7+x_8)}{1+{\vartheta }_1{x}_4+{\vartheta }_{12}{x}_7+{\vartheta }_{13}{x}_8}+\frac{{\beta }_2^h{x}_{11}}{1+{\vartheta }_2{x}_{11}}\right)\left(+\frac{{\beta }_3^h{x}_{12}}{1+{\vartheta }_3{x}_{12}}+{\mu }_h\right)x_2\equiv f_2,$$

$$\frac{d{x}_3}{dt}={\theta }_2{x}_1-\left(\frac{{\beta }_1(x_4+x_7+x_8)}{1+{\vartheta }_1{x}_4+{\vartheta }_{12}{x}_7+{\vartheta }_{13}{x}_8}+(1-{\eta }_2)\frac{{\beta }_2^h{x}_{11}}{1+{\vartheta }_2{x}_{11}}\right)\left(+(1-{\eta }_3)\frac{{\beta }_3^h{x}_{12}}{1+{\vartheta }_3{x}_{12}}+{\mu }_h\right)x_3\equiv f_3,$$

$$\frac{d{x}_4}{dt}=\frac{{\beta }_1(x_4+x_7+x_8)}{1+{\vartheta }_1{x}_4+{\vartheta }_{12}{x}_7+{\vartheta }_{13}{x}_8}(x_1+(1-{\eta }_1)x_2+x_3+x_9)-\left({\varphi }_c+{\zeta }_c+{\mu }_h\right)x_4-{\delta }_1\frac{{\beta }_2^h{x}_{11}}{1+{\vartheta }_2{x}_{11}}{x}_4-{\delta }_2\frac{{\beta }_3^h{x}_{12}}{1+{\vartheta }_3{x}_{12}}{x}_4\equiv f_4$$$$\frac{d{x}_5}{dt}=\frac{{\beta }_2^h{x}_{11}}{1+{\vartheta }_2{x}_{11}}(x_1+x_2+(1-{\eta }_2)x_3+x_9)+\xi \frac{{\beta }_2^h{x}_{11}}{1+{\vartheta }_2{x}_{11}}{x}_6-\left({\varphi }_d+{\zeta }_d+{\mu }_h\right)x_5-{\delta }_3\frac{{\beta }_1(x_4+x_7+x_8)}{1+{\vartheta }_1{x}_4+{\vartheta }_{12}{x}_7+{\vartheta }_{13}{x}_8}{x}_5\equiv f_5,$$$$\frac{d{x}_6}{dt}=\frac{{\beta }_3^h{x}_{12}}{1+{\vartheta }_3{x}_{12}}(x_1+x_2+(1-{\eta }_3)x_3+x_9)-\left({\varphi }_z+{\zeta }_z+{\mu }_h\right)x_6-{\delta }_4\frac{{\beta }_1(x_4+x_7+x_8)}{1+{\vartheta }_1{x}_4+{\vartheta }_{12}{x}_7+{\vartheta }_{13}{x}_8}{x}_6-\xi \frac{{\beta }_2^h{x}_{11}}{1+{\vartheta }_2{x}_{11}}{x}_6\equiv f_6,$$$$\frac{d{x}_7}{dt}={\delta }_1\frac{{\beta }_2^h{x}_{11}}{1+{\vartheta }_2{x}_{11}}{x}_4+{\delta }_3\frac{{\beta }_1(x_4+x_7+x_8)}{1+{\vartheta }_1{x}_4+{\vartheta }_{12}{x}_7+{\vartheta }_{13}{x}_8}{x}_5-\left({\varphi }_{cd}+{\zeta }_{cd}+{\mu }_h\right)x_7\equiv f_7,$$$$\frac{d{x}_8}{dt}={\delta }_2\frac{{\beta }_3^h{x}_{12}}{1+{\vartheta }_3{x}_{12}}{x}_4+{\delta }_4\frac{{\beta }_1(x_4+x_7+x_8)}{1+{\vartheta }_1{x}_4+{\vartheta }_{12}{x}_7+{\vartheta }_{13}{x}_8}{x}_6-\left({\varphi }_{cz}+{\zeta }_c+{\mu }_h\right)x_8\equiv f_8,$$$$\frac{d{x}_9}{dt}={\zeta }_c{x}_4+{\zeta }_d{x}_5+{\zeta }_z{x}_6+{\zeta }_{cd}{x}_7+{\zeta }_{cz}{x}_8-\left(\frac{{\beta }_1{x}_4}{1+{\vartheta }_1{x}_4}+\frac{{\beta }_2^h{x}_{11}}{1+{\vartheta }_2{x}_{11}}+\frac{{\beta }_3^h{x}_{12}}{1+{\vartheta }_3{x}_{12}}+{\mu }_h\right)x_9\equiv f_9,$$$$\frac{d{x}_{10}}{dt}={\Lambda }_v-\left[\frac{{\beta }_2^v(x_5+x_7)}{1+{\vartheta }_2{x}_5+{\vartheta }_{12}{x}_7}+\frac{{\beta }_3^v(x_6+x_8)}{1+{\vartheta }_3{x}_6+{\vartheta }_{13}{x}_8}+{\mu }_v\right]x_{10}\equiv f_{10},$$$$\frac{d{x}_{11}}{dt}=\frac{{\beta }_2^v(x_5+x_7)}{1+{\vartheta }_2{x}_5+{\vartheta }_{12}{x}_7}{x}_{10}-{\mu }_v{x}_{11}\equiv f_{11},$$$$\frac{d{x}_{12}}{dt}=\frac{{\beta }_3^v(x_6+x_8)}{1+{\vartheta }_3{x}_6+{\vartheta }_{13}{x}_8}{x}_{10}-{\mu }_v{x}_{12}\equiv f_{12}.$$ (11)

Consider the case ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0c},{\mathcal{R}}_{0d},{\mathcal{R}}_{0z}\}=1$. Let the parameter ${\beta }_1$ (say) be chosen as a bifurcation parameter. Evaluating ${\beta }_1={\beta }_1^{\ast }$ from ${\mathcal{R}}_{0c}=1$ gives

$${\beta }_1={\beta }_1^{\ast }=\frac{{\mu }_h({\mu }_h+{\theta }_1+{\theta }_2)({\varphi }_c+{\zeta }_c+{\mu }_h)}{{\Lambda }_h[{\mu }_h+(1-{\eta }_1){\theta }_1+{\theta }_2]}.$$

If Jacobian $\mathcal{J}$ of system (11) is evaluated at DFE $\left({\mathcal{G}}_0\right)$, then we get the matrix (see Box II).

where, $\mathcal{H}=({\mu }_h+{\theta }_1+{\theta }_2),{\beta }_{11}=(1-{\eta }_1){\beta }_1,{\beta }_{22}^h=(1-{\eta }_2){\beta }_2^h,{\beta }_{33}^h=(1-{\eta }_3){\beta }_3^h$,

${\mathcal{A}}^{\ast }=x_1^{\ast }+(1-{\eta }_1)x_2^{\ast }+x_3^{\ast }$, ${\mathcal{B}}^{\ast }=x_1^{\ast }+x_2^{\ast }+(1-{\eta }_2)x_3^{\ast }$, ${\mathcal{C}}^{\ast }=x_1^{\ast }+x_2^{\ast }+(1-{\eta }_3)x_3^{\ast }$.

The matrix $\mathcal{J}\left({\mathcal{G}}_0\right)$ will possess a right eigenvector (linked with zero eigenvalue of $J\left({\mathcal{G}}_0\right)$) given by $\omega ={\left[{\omega }_1,{\omega }_2,{\omega }_3,\dots ,{\omega }_{12}\right]}^T$, where

$${\omega }_1=-\frac{1}{({\mu }_h+{\theta }_1+{\theta }_2)}\left[{\beta }_1{x}_1^{\ast }{\omega }_4+{\beta }_2^h{x}_1^{\ast }{\omega }_{11}+{\beta }_3^h{x}_1^{\ast }{\omega }_{12}\right]<0,$$

$${\omega }_2=-\frac{1}{{\mu }_h}\left[-{\theta }_1{\omega }_1+{\beta }_1{x}_2^{\ast }{\omega }_4+(1-{\eta }_2){\beta }_2^h{x}_2^{\ast }{\omega }_{11}+{\beta }_3^h{x}_1^{\ast }{\omega }_{12}\right]<0,$$

$${\omega }_3=-\frac{1}{{\mu }_h}\left[-{\theta }_2{\omega }_1+{\beta }_1{x}_3^{\ast }{\omega }_4+{\beta }_2^h{x}_3^{\ast }{\omega }_{11}+(1-{\eta }_3){\beta }_3^h{x}_3^{\ast }{\omega }_{12}\right]<0,$$

$${\omega }_4=\frac{{\beta }_1{\mathcal{A}}^{\ast }}{{\mathcal{K}}_1}>0,$$

$${\omega }_5=\frac{{\beta }_2^h{\mathcal{B}}^{\ast }}{{\mu }_v{\mathcal{K}}_2}>0,{\omega }_6=\frac{{\beta }_3^h{\mathcal{C}}^{\ast }}{{\mu }_v{\mathcal{K}}_3}>0,{\omega }_7={\omega }_8=0,$$

$${\omega }_9=-\frac{1}{{\mu }_h}({\zeta }_c{\omega }_4+{\zeta }_d{\omega }_5+{\zeta }_z{\omega }_6)>0,$$

$${\omega }_{10}=-\frac{1}{{\mu }_v}({\beta }_2^v{x}_{10}^{\ast }{\omega }_5+{\beta }_3^v{x}_{10}^{\ast }{\omega }_6)<0,{\omega }_{11}=\frac{{\mathcal{K}}_2}{{\beta }_2^h{\beta }_2^v{x}_{10}^{\ast }{\mathcal{B}}^{\ast }}>0,$$

$${\omega }_{12}=\frac{{\mathcal{K}}_3}{{\beta }_3^h{\beta }_3^v{x}_{10}^{\ast }{\mathcal{C}}^{\ast }}>0.$$

The components of the left eigenvector of $\mathcal{J}\left({\mathcal{G}}_0\right){|}_{{\beta }_3={\beta }_3^{\ast }}$, $\nu =[{\nu }_1,{\nu }_2,\dots ,{\nu }_{12}]$ satisfying $\omega .\nu =1$ are

$${\nu }_1={\nu }_2={\nu }_3=0,{\nu }_4=\frac{{\beta }_1{\mathcal{A}}^{\ast }}{{\mathcal{K}}_1}>0,{\nu }_5=\frac{{\beta }_2^v{x}_{10}^{\ast }}{{\mu }_v{\mathcal{K}}_2}>0,{\nu }_6=\frac{{\beta }_3^v{x}_{10}^{\ast }}{{\mu }_v{\mathcal{K}}_3}>0,$$

$${\nu }_7=\frac{1}{{\mathcal{K}}_4}\left({\beta }_1{\mathcal{A}}^{\ast }{\nu }_4+{\beta }_2^v{x}_{10}^{\ast }{\nu }_{11}\right)>0,$$

$${\nu }_8=\frac{1}{{\mathcal{K}}_5}\left({\beta }_1{\mathcal{A}}^{\ast }{\nu }_4+{\beta }_3^v{x}_{10}^{\ast }{\nu }_{11}\right)>0,{\nu }_9={\nu }_{10}=0,{\nu }_{11}=\frac{{\mathcal{K}}_2}{{\beta }_2^h{\beta }_2^v{\mathcal{B}}^{\ast }{x}_{10}^{\ast }},$$

$${\nu }_{12}=\frac{{\mathcal{K}}_3}{{\beta }_3^h{\beta }_3^v{\mathcal{C}}^{\ast }{x}_{10}^{\ast }}$$

Adopting Theorem 4.1 [58] and by computing the non-zero partial derivatives of $f\left(x\right)$ (evaluated at the disease free equilibrium, (${\mathcal{G}}_0$)), the associated bifurcation coefficients are given below

$$a=\sum _{k,i,j=1}^{12}{\nu }_k{\omega }_i{\omega }_j\frac{{\partial }^2{f}_k}{\partial x_i\partial x_j}(0,0)\text{and}b=\sum _{k,i=1}^{12}{\nu }_k{\omega }_i\frac{{\partial }^2{f}_k}{\partial x_i\partial {\beta }_1^{\ast }}(0,0),$$

where,

$$a=2{\omega }_4{\nu }_4\text{{}{\beta }_1[{\omega }_1+(1-{\eta }_1){\omega }_2+{\omega }_3-2{\vartheta }_1{\mathcal{A}}^{\ast }{\omega }_4+{\omega }_9]-[{\delta }_1{\beta }_2^h{\omega }_{11}+{\delta }_2{\beta }_3^h{\omega }_{11}]\text{}}+2{\beta }_2^h{\omega }_{11}{\nu }_5\{{\omega }_1+{\omega }_2+(1-{\eta }_2){\omega }_3-2{\vartheta }_2{\mathcal{B}}^{\ast }{\omega }_{11}+\xi {\omega }_6+{\omega }_9\}-2{\beta }_1{\delta }_3{\omega }_4{\omega }_5{\nu }_5$$

$$+2{\beta }_3^h{\omega }_{12}{\nu }_6\{{\omega }_1+{\omega }_2+(1-{\eta }_3){\omega }_3-2{\vartheta }_3{\mathcal{C}}^{\ast }{\omega }_{12}+{\omega }_9\}-2{\beta }_1{\delta }_4{\omega }_4{\omega }_6{\nu }_6-\xi {\beta }_2^h{\omega }_6{\omega }_{11}{\nu }_6-2{\beta }_2^v({\omega }_5+{\omega }_7){\nu }_{11}({\vartheta }_2{\omega }_5{x}_{10}^{\ast }+{\vartheta }_{12}{\omega }_7{x}_{10}^{\ast }-{\omega }_{10})-2{\beta }_3^v({\omega }_6+{\omega }_8){\nu }_{12}({\vartheta }_2{\omega }_6{x}_{10}^{\ast }+{\vartheta }_{13}{\omega }_8{x}_{10}^{\ast }-{\omega }_{10})+2{\omega }_4{\nu }_7({\delta }_1{\beta }_2^h{\omega }_{11}+{\delta }_3{\beta }_1{\omega }_5)+2{\omega }_4{\nu }_8({\delta }_2{\beta }_2^h{\omega }_{12}+{\delta }_4{\beta }_1{\omega }_6)$$ (12)

$$b=\sum _{k,i=1}^{12}{\nu }_k{\omega }_i\frac{{\partial }^2{f}_k}{\partial x_i\partial {\beta }_1^{\ast }}(0,0)=[x_1^{\ast }+(1-{\eta }_1)x_2^{\ast }+x_3^{\ast }]{\omega }_4{\nu }_4>0.$$ (13)

Based on Theorem 4.1 in [58], system (1) will exhibit backward bifurcation as long as the coefficient $a>0$.

It is interesting to note that, if we assume no re-infection with same or different disease, and set the super-infection and co-infection related parameters to zero, that is, $\xi ={\delta }_1={\delta }_2={\delta }_3={\delta }_4=0$, then the co-efficient,

$$a=2{\omega }_4{\nu }_4\left\{{\beta }_1[{\omega }_1+(1-{\eta }_1){\omega }_2+{\omega }_3-2{\vartheta }_1{\mathcal{A}}^{\ast }{\omega }_4]\right\}+2{\beta }_2^h{\omega }_{11}{\nu }_5\text{{}{\omega }_1+{\omega }_2+(1-{\eta }_2){\omega }_3-2{\vartheta }_2{\mathcal{B}}^{\ast }{\omega }_{11}\text{}}+2{\beta }_3^h{\omega }_{12}{\nu }_6\{{\omega }_1+{\omega }_2+(1-{\eta }_3){\omega }_3-2{\vartheta }_3{\mathcal{C}}^{\ast }{\omega }_{12}\}-2{\beta }_2^v({\omega }_5+{\omega }_7){\nu }_{11}({\vartheta }_2{\omega }_5{x}_{10}^{\ast }+{\vartheta }_{12}{\omega }_7{x}_{10}^{\ast }-{\omega }_{10})-2{\beta }_3^v({\omega }_6+{\omega }_8){\nu }_{12}({\vartheta }_2{\omega }_6{x}_{10}^{\ast }+{\vartheta }_{13}{\omega }_8{x}_{10}^{\ast }-{\omega }_{10})<0,\text{(since}{\omega }_1<0,$$

$${\omega }_2<0,{\omega }_3<0,{\omega }_{10}<0\text{)}.$$

Thus, ruling out the possibility of backward bifurcation in the model (1).

Theorem 3.5 under the condition that $a>0$ implies that the control of the triple infections becomes very difficult, even when the reproduction number is less than one. However, if we rule out re-infection with same or a different disease, co-infection and super-infection, such that $a<0$, then the control of the triple diseases becomes feasible in the population.

Box II.

$$\left(\begin{array}{cccccccccccc}\hfill -\mathcal{H}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_1{x}_1^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_1{x}_1^{\ast }\hfill & \hfill -{\beta }_1{x}_1^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_2^h{x}_1^{\ast }\hfill & \hfill -{\beta }_3^h{x}_1^{\ast }\hfill \\ \hfill {\theta }_1\hfill & \hfill -{\mu }_h\hfill & \hfill 0\hfill & \hfill -{\beta }_{11}{x}_2^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_{11}{x}_2^{\ast }\hfill & \hfill -{\beta }_{11}{x}_2^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_2^h{x}_2^{\ast }\hfill & \hfill -{\beta }_3^h{x}_2^{\ast }\hfill \\ \hfill {\theta }_2\hfill & \hfill 0\hfill & \hfill -{\mu }_h\hfill & \hfill -{\beta }_1{x}_3^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_1{x}_3^{\ast }\hfill & \hfill -{\beta }_1{x}_3^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_{22}^h{x}_3^{\ast }\hfill & \hfill -{\beta }_{33}^h{x}_3^{\ast }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_1{\mathcal{A}}^{\ast }-{\mathcal{K}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_1{\mathcal{A}}^{\ast }\hfill & \hfill {\beta }_1{\mathcal{A}}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mathcal{K}}_2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_2^h{\mathcal{B}}^{\ast }\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mathcal{K}}_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_3^h{\mathcal{C}}^{\ast }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mathcal{K}}_4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mathcal{K}}_5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\zeta }_c\hfill & \hfill {\zeta }_d\hfill & \hfill {\zeta }_z\hfill & \hfill {\zeta }_{cd}\hfill & \hfill {\zeta }_{cz}\hfill & \hfill -{\mu }_h\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_2^v{x}_{10}^{\ast }\hfill & \hfill -{\beta }_3^v{x}_{10}^{\ast }\hfill & \hfill -{\beta }_2^v{x}_{10}^{\ast }\hfill & \hfill -{\beta }_3^v{x}_{10}^{\ast }\hfill & \hfill 0\hfill & \hfill -{\mu }_v\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_2^v{x}_{10}^{\ast }\hfill & \hfill 0\hfill & \hfill {\beta }_2^v{x}_{10}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mu }_v\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_3^v{x}_{10}^{\ast }\hfill & \hfill 0\hfill & \hfill {\beta }_3^v{x}_{10}^{\ast }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mu }_v\hfill \end{array}\right),$$

3.6. Global asymptotic stability (GAS) of disease free and endemic equilibria of the model (1)

In order to establish the global stability of an epidemic system, one of the most effective methods is the direct Lyapunov method [59] which requires an auxiliary function $\mathcal{L}\left(X\right)$ (say), with $X\in {\mathbb{R}}^n$, defined on a neighborhood $U$ of the origin, 0, and satisfies the following properties:

• (i.)$\mathcal{L}\left(X\right)>0$, for all $X\in U\setminus \left\{\mathbf{0}\right\}$,
• (ii.)$\mathcal{L}\left(\mathbf{0}\right)=0$,
• (iii.)$\frac{d\mathcal{L}}{dt}\le 0$, for all $X\in U$.

3.6.1. Global asymptotic stability of disease free equilibrium of the model (1), for a special case

By setting the cause of backward bifurcation to zero, and with the help of appropriately constructed Lyapunov function, we shall prove the stability of disease free equilibrium.

Theorem 3.5

Assuming no co-infection or super-infection in the model (1) , (that is, $\xi ={\delta }_1={\delta }_2={\delta }_3={\delta }_4=0$ ), the model’s DFE given by ${\mathcal{Q}}_0$ , is GAS in $\mathcal{D}$ given that ${\mathcal{R}}_0\le 1$ .

Proof

Consider the reduced model without co-infection or super-infection.

$$\frac{d{\mathcal{S}}_h\left(t\right)}{dt}={\Lambda }_h-\left(\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}+\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}+\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}+{\mu }_h+{\theta }_1+{\theta }_2\right){\mathcal{S}}_h,$$$$\frac{d{\mathcal{S}}_{hc}\left(t\right)}{dt}={\theta }_1{\mathcal{S}}_h-\left((1-{\eta }_1)\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}+\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}+\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}+{\mu }_h\right){\mathcal{S}}_{hc},$$$$\frac{d{\mathcal{S}}_{hd}\left(t\right)}{dt}={\theta }_2{\mathcal{S}}_h-\left(\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}+(1-{\eta }_2)\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}+(1-{\eta }_3)\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}\right)\left(+{\mu }_h\right){\mathcal{S}}_{hd},$$$$\frac{d{\mathcal{I}}_c\left(t\right)}{dt}=\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}({\mathcal{S}}_h+(1-{\eta }_1){\mathcal{S}}_{hc}+{\mathcal{S}}_{hd}+\mathcal{R})-\left({\varphi }_c+{\zeta }_c+{\mu }_h\right){\mathcal{I}}_c,$$$$\frac{d{\mathcal{I}}_d\left(t\right)}{dt}=\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_2){\mathcal{S}}_{hd}+\mathcal{R})-\left({\varphi }_d+{\zeta }_d+{\mu }_h\right){\mathcal{I}}_d,$$$$\frac{d{\mathcal{I}}_z\left(t\right)}{dt}=\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_3){\mathcal{S}}_{hd}+\mathcal{R})-\left({\varphi }_z+{\zeta }_z+{\mu }_h\right){\mathcal{I}}_z,$$$$\frac{d\mathcal{R}\left(t\right)}{dt}={\zeta }_c{\mathcal{I}}_c+{\zeta }_d{\mathcal{I}}_d+{\zeta }_z{\mathcal{I}}_z-\left({\mu }_h+\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}+\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}+\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}\right)\mathcal{R},$$$$\frac{d{\mathcal{S}}_v\left(t\right)}{dt}={\Lambda }_v-\left[\frac{{\beta }_2^v{\mathcal{I}}_d}{1+{\vartheta }_2{\mathcal{I}}_d}+\frac{{\beta }_3^v{\mathcal{I}}_z}{1+{\vartheta }_3{\mathcal{I}}_z}+{\mu }_v\right]{\mathcal{S}}_v,$$$$\frac{d{\mathcal{I}}_d^v\left(t\right)}{dt}=\frac{{\beta }_2^v{\mathcal{I}}_d}{1+{\vartheta }_2{\mathcal{I}}_d}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_d^v,$$$$\frac{d{\mathcal{I}}_z^v\left(t\right)}{dt}=\frac{{\beta }_3^v{\mathcal{I}}_z}{1+{\vartheta }_3{\mathcal{I}}_z}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_z^v,$$ (14)

Consider the Lyapunov candidate:

$$\mathcal{L}=\frac{1}{{\mathcal{K}}_1}{\mathcal{I}}_c+\frac{{\beta }_2^v{\mathcal{S}}_v^{\ast }}{{\mu }_v{\mathcal{K}}_2}{\mathcal{I}}_d+\frac{{\beta }_3^v{\mathcal{S}}_v^{\ast }}{{\mu }_v{\mathcal{K}}_3}{\mathcal{I}}_z+\frac{{\mathcal{R}}_{0d}}{{\mu }_v}{\mathcal{I}}_d^v+\frac{{\mathcal{R}}_{0z}}{{\mu }_v}{\mathcal{I}}_z^v,$$

with time derivative

$${\dot{\mathcal{L}}}_1=\frac{{\beta }_2^v{\mathcal{S}}_v^{\ast }}{{\mu }_v{\mathcal{K}}_2}\left(\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_2){\mathcal{S}}_{hd})-\left({\varphi }_d+{\zeta }_d+{\mu }_h\right){\mathcal{I}}_d\right)+\frac{{\mathcal{R}}_{0d}}{{\mu }_v}\left(\frac{{\beta }_2^v{\mathcal{I}}_d}{1+{\vartheta }_2{\mathcal{I}}_d}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_d^v\right)+\frac{{\beta }_3^v{\mathcal{S}}_v^{\ast }}{{\mu }_v{\mathcal{K}}_3}\left(\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_3){\mathcal{S}}_{hd})-\left({\varphi }_z+{\zeta }_z+{\mu }_h\right){\mathcal{I}}_z\right)+\frac{{\mathcal{R}}_{0z}}{{\mu }_v}\left(\frac{{\beta }_3^v{\mathcal{I}}_z}{1+{\vartheta }_3{\mathcal{I}}_z}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_z^v\right),$$

which can be further simplified into

$${\dot{\mathcal{L}}}_1=\frac{1}{{\mathcal{K}}_1}\left(\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}({\mathcal{S}}_h+(1-{\eta }_1){\mathcal{S}}_{hc}+{\mathcal{S}}_{hd}+\mathcal{R})-{\mathcal{K}}_1{\mathcal{I}}_c\right)+\frac{{\beta }_2^v{\mathcal{S}}_v^{\ast }}{{\mu }_v{\mathcal{K}}_2}\left(\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_2){\mathcal{S}}_{hd}+\mathcal{R})-{\mathcal{K}}_2{\mathcal{I}}_d\right)$$

$$+\frac{{\beta }_3^v{\mathcal{S}}_v^{\ast }}{{\mu }_v{\mathcal{K}}_3}\left(\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_3){\mathcal{S}}_{hd}+\mathcal{R})-{\mathcal{K}}_3{\mathcal{I}}_z\right)+\frac{{\mathcal{R}}_{0d}}{{\mu }_v}\left(\frac{{\beta }_2^v{\mathcal{I}}_d}{1+{\vartheta }_2{\mathcal{I}}_d}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_d^v\right)+\frac{{\mathcal{R}}_{0z}}{{\mu }_v}\left(\frac{{\beta }_3^v{\mathcal{I}}_z}{1+{\vartheta }_3{\mathcal{I}}_z}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_z^v\right),$$

Simplifying further (noting that ${\mathcal{S}}_h+{\mathcal{S}}_{hc}+{\mathcal{S}}_{hd}+{\mathcal{I}}_c+{\mathcal{I}}_d+{\mathcal{I}}_z+{\mathcal{I}}_{cd}+{\mathcal{I}}_{cz}+\mathcal{R}\le \frac{{\Lambda }_h}{{\mu }_h},{\mathcal{S}}_v+{\mathcal{I}}_d^v+{\mathcal{I}}_z^v\le \frac{{\Lambda }_v}{{\mu }_v}$, and ${\mathcal{S}}_v<{\mathcal{S}}_v^{\ast }$), we have

$${\dot{\mathcal{L}}}_1\le \frac{1}{{\mathcal{K}}_1}\left({\beta }_1{\mathcal{I}}_c({\mathcal{S}}_h^{\ast }+(1-{\eta }_1){\mathcal{S}}_{hc}^{\ast }+{\mathcal{S}}_{hd}^{\ast })-{\mathcal{K}}_1{\mathcal{I}}_c\right)+\frac{{\beta }_2^v{\mathcal{S}}_v^{\ast }}{{\mu }_v{\mathcal{K}}_2}\left({\beta }_2^h{\mathcal{I}}_d^v({\mathcal{S}}_h^{\ast }+{\mathcal{S}}_{hc}^{\ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast })-{\mathcal{K}}_2{\mathcal{I}}_d\right)+\frac{{\beta }_3^v{\mathcal{S}}_v^{\ast }}{{\mu }_v{\mathcal{K}}_3}\left({\beta }_3^h{\mathcal{I}}_z^v({\mathcal{S}}_h^{\ast }+{\mathcal{S}}_{hc}^{\ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast })-{\mathcal{K}}_3{\mathcal{I}}_z\right)+\frac{{\mathcal{R}}_{0d}}{{\mu }_v}\left({\beta }_2^v{\mathcal{I}}_d{\mathcal{S}}_v^{\ast }-{\mu }_v{\mathcal{I}}_d^v\right)+\frac{{\mathcal{R}}_{0z}}{{\mu }_v}\left({\beta }_3^v{\mathcal{I}}_z{\mathcal{S}}_v^{\ast }-{\mu }_v{\mathcal{I}}_z^v\right),=\left(\frac{{\beta }_1{\Lambda }_h[{\mu }_h+(1-{\eta }_1){\theta }_1+{\theta }_2]}{{\mu }_h({\mu }_h+{\theta }_1+{\theta }_2)({\varphi }_c+{\zeta }_c+{\mu }_h)}-1\right){\mathcal{I}}_c+\left(\frac{{\beta }_2^h{\beta }_2^v{\Lambda }_h{\Lambda }_v[{\mu }_h+{\theta }_1+(1-{\eta }_2){\theta }_2]}{{\mu }_h{\mu }_v^2({\mu }_h+{\theta }_1+{\theta }_2)({\varphi }_d+{\zeta }_d+{\mu }_h)}-{\mathcal{R}}_{0d}\right){\mathcal{I}}_d^v+\left(\frac{{\beta }_3^h{\beta }_3^v{\Lambda }_h{\Lambda }_v[{\mu }_h+{\theta }_1+(1-{\eta }_3){\theta }_2]}{{\mu }_h{\mu }_v^2({\mu }_h+{\theta }_1+{\theta }_2)({\varphi }_z+{\zeta }_z+{\mu }_h)}-{\mathcal{R}}_{0z}\right){\mathcal{I}}_z^v+\frac{{\beta }_2^v{\mathcal{S}}_v^{\ast }}{{\mu }_v}\left({\mathcal{R}}_{0d}-1\right){\mathcal{I}}_d+\frac{{\beta }_3^v{\mathcal{S}}_v^{\ast }}{{\mu }_v}\left({\mathcal{R}}_{0z}-1\right){\mathcal{I}}_z=({\mathcal{R}}_{0c}-1){\mathcal{I}}_c+\frac{{\beta }_2^v{\mathcal{S}}_v^{\ast }}{{\mu }_v}({\mathcal{R}}_{0d}-1){\mathcal{I}}_d+{\mathcal{R}}_{0d}({\mathcal{R}}_{0d}-1){\mathcal{I}}_d^v+\frac{{\beta }_3^v{\mathcal{S}}_v^{\ast }}{{\mu }_v}({\mathcal{R}}_{0z}-1){\mathcal{I}}_z+{\mathcal{R}}_{0z}({\mathcal{R}}_{0z}-1){\mathcal{I}}_z^v$$

Since all the model parameters and variables are non-negative, it follows that ${\dot{\mathcal{L}}}_1\le 0$ for ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0c},{\mathcal{R}}_{0d},{\mathcal{R}}_{0z}\}\le 1$, and ${\dot{\mathcal{L}}}_1=0$ if and only if ${\mathcal{I}}_c={\mathcal{I}}_d={\mathcal{I}}_z={\mathcal{I}}_{cd}={\mathcal{I}}_{cz}={\mathcal{I}}_d^v={\mathcal{I}}_z^v=0$. Thus, ${\mathcal{L}}_1$ is one of the candidates of a Lyapunov function on $\mathcal{D}$. By La Salle’s Invariance Principle [59], ${\mathcal{I}}_c\to 0,{\mathcal{I}}_d\to 0,{\mathcal{I}}_z\to 0,{\mathcal{I}}_{cd}\to 0,{\mathcal{I}}_{cz}\to 0,{\mathcal{I}}_d^v\to 0\text{and}{\mathcal{I}}_z^v\to 0$ as $t\to \infty$. Substituting ${\mathcal{I}}_c={\mathcal{I}}_d={\mathcal{I}}_z={\mathcal{I}}_{cd}={\mathcal{I}}_{cz}={\mathcal{I}}_d^v={\mathcal{I}}_z^v=0$ in (1) results in $\mathcal{R}\to 0,{\mathcal{S}}_h\to {\mathcal{S}}_h^{\ast }$, ${\mathcal{S}}_v\to {\mathcal{S}}_v^{\ast }$ as $t\to \infty$. Thus, it is concluded that every solution to system (1) with $\xi ={\delta }_1={\delta }_2={\delta }_3={\delta }_4=0$, having initial conditions in $\mathcal{D}$, converges to the DFE as $t\to \infty$ whenever ${\mathcal{R}}_0\le 1$. Epidemiologically, the above result states; in the absence of co-infection and super-infection, all three diseases can be eliminated if ${\mathcal{R}}_0\le 1$, regardless of the initial sizes of the sub-populations.

3.6.2. Global asymptotic stability of endemic equilibrium of the model (1), for a special case

Theorem 3.6

Assuming absence of re-infection with the same or a different disease, and absence of co-infection and super-infection in the model (1) (that is, ${\delta }_1={\delta }_2={\delta }_3={\delta }_4=\xi =0$ ), the model’s endemic equilibrium given by ${\mathcal{Q}}_e$ , is GAS in $\mathcal{D}$ given that ${\mathcal{R}}_0>1$ .

Proof

Consider the candidate for a Lyapunov function defined below (whose type has been successfully used to establish stability of endemic equilibrium for disease models [60], [61]):

$${\mathcal{L}}_2={\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}[{\mathcal{S}}_h-{\mathcal{S}}_h^{\ast \ast }-{\mathcal{S}}_h^{\ast \ast }ln\left(\frac{{\mathcal{S}}_h}{{\mathcal{S}}_h^{\ast \ast }}\right)+{\mathcal{S}}_{hc}-{\mathcal{S}}_{hc}^{\ast \ast }-{\mathcal{S}}_{hc}^{\ast \ast }ln\left(\frac{{\mathcal{S}}_{hc}}{{\mathcal{S}}_{hc}^{\ast \ast }}\right)+{\mathcal{S}}_{hd}-{\mathcal{S}}_{hd}^{\ast \ast }-{\mathcal{S}}_{hd}^{\ast \ast }ln\left(\frac{{\mathcal{S}}_{hd}}{{\mathcal{S}}_{hd}^{\ast \ast }}\right)+{\mathcal{I}}_c-{\mathcal{I}}_c^{\ast \ast }-{\mathcal{I}}_c^{\ast \ast }ln\left(\frac{{\mathcal{I}}_c}{{\mathcal{I}}_c^{\ast \ast }}\right)+{\mathcal{I}}_z-{\mathcal{I}}_z^{\ast \ast }-{\mathcal{I}}_z^{\ast \ast }ln\left(\frac{{\mathcal{I}}_z}{{\mathcal{I}}_z^{\ast \ast }}\right)+{\mathcal{I}}_d-{\mathcal{I}}_d^{\ast \ast }-{\mathcal{I}}_d^{\ast \ast }ln\left(\frac{{\mathcal{I}}_d}{{\mathcal{I}}_d^{\ast \ast }}\right)]+{\beta }_2^h{\beta }_3^h[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }]\text{[}{\mathcal{S}}_h^{\ast \ast }$$

$$+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }\text{]}\times [{\mathcal{S}}_v-{\mathcal{S}}_v^{\ast \ast }-{\mathcal{S}}_v^{\ast \ast }ln\left(\frac{{\mathcal{S}}_v}{{\mathcal{S}}_v^{\ast \ast }}\right)+{\mathcal{I}}_d^v-{\mathcal{I}}_d^{v\ast \ast }-{\mathcal{I}}_d^{v\ast \ast }ln\left(\frac{{\mathcal{I}}_d^v}{{\mathcal{I}}_d^{v\ast \ast }}\right)+{\mathcal{I}}_z^v-{\mathcal{I}}_z^{v\ast \ast }-{\mathcal{I}}_z^{v\ast \ast }ln\left(\frac{{\mathcal{I}}_z^v}{{\mathcal{I}}_z^{v\ast \ast }}\right)]$$

with Lyapunov derivative,

$${\dot{\mathcal{L}}}_2={\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}[\left(1-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}\right)\dot{{\mathcal{S}}_h}+\left(1-\frac{{\mathcal{S}}_{hc}^{\ast \ast }}{{\mathcal{S}}_{hc}}\right)\dot{{\mathcal{S}}_{hc}}+\left(1-\frac{{\mathcal{S}}_{hd}^{\ast \ast }}{{\mathcal{S}}_{hd}}\right)\dot{{\mathcal{S}}_{hd}}+\left(1-\frac{{\mathcal{I}}_c^{\ast \ast }}{{\mathcal{I}}_c}\right){\dot{\mathcal{I}}}_c+\left(1-\frac{{\mathcal{I}}_z^{\ast \ast }}{{\mathcal{I}}_z}\right){\dot{\mathcal{I}}}_z+\left(1-\frac{{\mathcal{I}}_d^{\ast \ast }}{{\mathcal{I}}_d}\right){\dot{\mathcal{I}}}_d]+{\beta }_2^h{\beta }_3^h[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }]\text{[}{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }\text{]}\left[\left(1-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}\right)\dot{{\mathcal{S}}_v}+\left(1-\frac{{\mathcal{I}}_d^{v\ast \ast }}{{\mathcal{I}}_d^v}\right){\dot{\mathcal{I}}}_d^v\right]$$ (15)

Substituting the derivatives in (1) into ${\dot{\mathcal{L}}}_2$, we have

$${\dot{\mathcal{L}}}_2={\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}\right)\left[{\Lambda }_h-\left(\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}+\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}+\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}\right)\right)$$$$\left(\left(+{\mu }_h+{\theta }_1+{\theta }_2\right){\mathcal{S}}_h\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{S}}_{hc}^{\ast \ast }}{{\mathcal{S}}_{hc}}\right)\left[{\theta }_1{\mathcal{S}}_h-\left((1-{\eta }_1)\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}\right)\right)$$$$\left(\left(+\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}+\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}+{\mu }_h\right){\mathcal{S}}_{hc}\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{S}}_{hd}^{\ast \ast }}{{\mathcal{S}}_{hd}}\right)\left[{\theta }_2{\mathcal{S}}_h-\left(\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}+(1-{\eta }_2)\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}\right)\right)$$$$\left(\left(+(1-{\eta }_3)\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}+{\mu }_h\right){\mathcal{S}}_{hd}\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{I}}_c^{\ast \ast }}{{\mathcal{I}}_c}\right)\left[\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}({\mathcal{S}}_h+(1-{\eta }_1){\mathcal{S}}_{hc}+{\mathcal{S}}_{hd})\right)\left(-\left({\varphi }_c+{\zeta }_c+{\mu }_h\right){\mathcal{I}}_c\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{I}}_d^{\ast \ast }}{{\mathcal{I}}_d}\right)\left[\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_2){\mathcal{S}}_{hd})\right).\left(-\left({\varphi }_d+{\zeta }_d+{\mu }_h\right){\mathcal{I}}_d\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{I}}_z^{\ast \ast }}{{\mathcal{I}}_z}\right)\left[\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_3){\mathcal{S}}_{hd})\right)\left(-\left({\varphi }_z+{\zeta }_z+{\mu }_h\right){\mathcal{I}}_z\right]+{\beta }_2^h{\beta }_3^h[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }][{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }]$$$$\left(1-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}\right)\left[{\Lambda }_v-\left[\frac{{\beta }_2^v{\mathcal{I}}_d}{1+{\vartheta }_2{\mathcal{I}}_d}+\frac{{\beta }_3^v{\mathcal{I}}_z}{1+{\vartheta }_3{\mathcal{I}}_z}+{\mu }_v\right]{\mathcal{S}}_v\right]+{\beta }_2^h{\beta }_3^h[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }][{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }]$$$$\left(1-\frac{{\mathcal{I}}_d^{v\ast \ast }}{{\mathcal{I}}_d^v}\right)\left[\frac{{\beta }_2^v{\mathcal{I}}_d}{1+{\vartheta }_2{\mathcal{I}}_d}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_d^v\right]+{\beta }_3^h{\mathcal{S}}_h^{\ast \ast }\left(1-\frac{{\mathcal{I}}_z^{v\ast \ast }}{{\mathcal{I}}_z^v}\right)\left[\frac{{\beta }_3^v{\mathcal{I}}_z}{1+{\vartheta }_3{\mathcal{I}}_z}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_z^v\right]$$ (16)

From model (1) at steady state, we have

$${\Lambda }_h=\left(\frac{{\beta }_1{\mathcal{I}}_c^{\ast \ast }}{1+{\vartheta }_1{\mathcal{I}}_c^{\ast \ast }}+\frac{{\beta }_2^h{\mathcal{I}}_d^{v\ast \ast }}{1+{\vartheta }_2{\mathcal{I}}_d^{v\ast \ast }}+\frac{{\beta }_3^h{\mathcal{I}}_z^{v\ast \ast }}{1+{\vartheta }_3{\mathcal{I}}_z^{v\ast \ast }}+{\mu }_h+{\theta }_1+{\theta }_2\right){\mathcal{S}}_h^{\ast \ast },$$

$${\theta }_1{\mathcal{S}}_h^{\ast \ast }=\left((1-{\eta }_1)\frac{{\beta }_1{\mathcal{I}}_c^{\ast \ast }}{1+{\vartheta }_1{\mathcal{I}}_c^{\ast \ast }}+\frac{{\beta }_2^h{\mathcal{I}}_d^{v\ast \ast }}{1+{\vartheta }_2{\mathcal{I}}_d^{v\ast \ast }}+\frac{{\beta }_3^h{\mathcal{I}}_z^{v\ast \ast }}{1+{\vartheta }_3{\mathcal{I}}_z^{v\ast \ast }}+{\mu }_h\right){\mathcal{S}}_{hc}^{\ast \ast },$$

$${\theta }_2{\mathcal{S}}_h^{\ast \ast }=\left(\frac{{\beta }_1{\mathcal{I}}_c^{\ast \ast }}{1+{\vartheta }_1{\mathcal{I}}_c^{\ast \ast }}+(1-{\eta }_2)\frac{{\beta }_2^h{\mathcal{I}}_d^{v\ast \ast }}{1+{\vartheta }_2{\mathcal{I}}_d^{v\ast \ast }}+(1-{\eta }_3)\frac{{\beta }_3^h{\mathcal{I}}_z^{v\ast \ast }}{1+{\vartheta }_3{\mathcal{I}}_z^{v\ast \ast }}+{\mu }_h\right){\mathcal{S}}_{hd}^{\ast \ast },$$$$\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c^{\ast \ast }}\text{(}{\mathcal{S}}_h^{\ast \ast }+(1-{\eta }_1){\mathcal{S}}_{hc}^{\ast \ast }+{\mathcal{S}}_{hd}^{\ast \ast }\text{)}=\left({\varphi }_c+{\zeta }_c+{\mu }_h\right){\mathcal{I}}_c^{\ast \ast },$$$$\frac{{\beta }_2^h{\mathcal{I}}_d^{v\ast \ast }}{1+{\vartheta }_2{\mathcal{I}}_d^{v\ast \ast }}\text{(}{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }\text{)}=\left({\varphi }_d+{\zeta }_d+{\mu }_h\right){\mathcal{I}}_d^{\ast \ast },$$$$\frac{{\beta }_3^h{\mathcal{I}}_z^{v\ast \ast }}{1+{\vartheta }_3{\mathcal{I}}_z^{v\ast \ast }}\text{(}{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }\text{)}=\left({\varphi }_z+{\zeta }_z+{\mu }_h\right){\mathcal{I}}_z^{\ast \ast },$$$${\zeta }_c{\mathcal{I}}_c^{\ast \ast }+{\zeta }_d{\mathcal{I}}_d^{\ast \ast }+{\zeta }_z{\mathcal{I}}_z^{\ast \ast }={\mu }_h{\mathcal{R}}^{\ast \ast },$$$${\Lambda }_v=\left[\frac{{\beta }_2^v{\mathcal{I}}_d^{\ast \ast }}{1+{\vartheta }_2{\mathcal{I}}_d^{\ast \ast }}+\frac{{\beta }_3^v{\mathcal{I}}_z^{\ast \ast }}{1+{\vartheta }_3{\mathcal{I}}_z^{\ast \ast }}+{\mu }_v\right]{\mathcal{S}}_v^{\ast \ast },$$$$\frac{{\beta }_2^v{\mathcal{I}}_d^{\ast \ast }}{1+{\vartheta }_2{\mathcal{I}}_d^{\ast \ast }}{\mathcal{S}}_v^{\ast \ast }={\mu }_v{\mathcal{I}}_d^{v\ast \ast },$$$$\frac{{\beta }_3^v{\mathcal{I}}_z^{\ast \ast }}{1+{\vartheta }_3{\mathcal{I}}_z^{\ast \ast }}{\mathcal{S}}_v^{\ast \ast }={\mu }_v{\mathcal{I}}_z^{v\ast \ast },$$ (17)

Substituting the expressions in (17) into (16), gives

$${\dot{\mathcal{L}}}_2={\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}\right)[\left(\frac{{\beta }_1{\mathcal{I}}_c^{\ast \ast }}{1+{\vartheta }_1{\mathcal{I}}_c^{\ast \ast }}+\frac{{\beta }_2^h{\mathcal{I}}_d^{v\ast \ast }}{1+{\vartheta }_2{\mathcal{I}}_d^{v\ast \ast }}+\frac{{\beta }_3^h{\mathcal{I}}_z^{v\ast \ast }}{1+{\vartheta }_3{\mathcal{I}}_z^{v\ast \ast }}\right)\left(+{\mu }_h+{\theta }_1+{\theta }_2\right){\mathcal{S}}_h^{\ast \ast }-\left(\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}+\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}+\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}+{\mu }_h+{\theta }_1+{\theta }_2\right){\mathcal{S}}_h]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{S}}_{hc}^{\ast \ast }}{{\mathcal{S}}_{hc}}\right)\left[{\theta }_1{\mathcal{S}}_h-\left((1-{\eta }_1)\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}+\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}\right)\right)$$

$$\left(\left(+\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}+{\mu }_h\right){\mathcal{S}}_{hc}\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{S}}_{hd}^{\ast \ast }}{{\mathcal{S}}_{hd}}\right)\left[{\theta }_2{\mathcal{S}}_h-\left(\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}+(1-{\eta }_2)\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}\right)\right)$$

$$\left(\left(+(1-{\eta }_3)\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}+{\mu }_h\right){\mathcal{S}}_{hd}\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{I}}_c^{\ast \ast }}{{\mathcal{I}}_c}\right)\left[\frac{{\beta }_1{\mathcal{I}}_c}{1+{\vartheta }_1{\mathcal{I}}_c}({\mathcal{S}}_h+(1-{\eta }_1){\mathcal{S}}_{hc}+{\mathcal{S}}_{hd})\right)\left(-\left({\varphi }_c+{\zeta }_c+{\mu }_h\right){\mathcal{I}}_c\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{I}}_d^{\ast \ast }}{{\mathcal{I}}_d}\right)\left[\frac{{\beta }_2^h{\mathcal{I}}_d^v}{1+{\vartheta }_2{\mathcal{I}}_d^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_2){\mathcal{S}}_{hd})\right)\left(-\left({\varphi }_d+{\zeta }_d+{\mu }_h\right){\mathcal{I}}_d\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{I}}_z^{\ast \ast }}{{\mathcal{I}}_z}\right)\left[\frac{{\beta }_3^h{\mathcal{I}}_z^v}{1+{\vartheta }_3{\mathcal{I}}_z^v}({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_3){\mathcal{S}}_{hd})\right)\left(-\left({\varphi }_z+{\zeta }_z+{\mu }_h\right){\mathcal{I}}_z\right]+{\beta }_2^h{\beta }_3^h[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }][{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }]\left(1-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}\right)\times [\left[\frac{{\beta }_2^v{\mathcal{I}}_d^{\ast \ast }}{1+{\vartheta }_2{\mathcal{I}}_d^{\ast \ast }}+\frac{{\beta }_3^v{\mathcal{I}}_z^{\ast \ast }}{1+{\vartheta }_3{\mathcal{I}}_z^{\ast \ast }}+{\mu }_v\right]{\mathcal{S}}_v^{\ast \ast }-\left[\frac{{\beta }_2^v{\mathcal{I}}_d}{1+{\vartheta }_2{\mathcal{I}}_d}+\frac{{\beta }_3^v{\mathcal{I}}_z}{1+{\vartheta }_3{\mathcal{I}}_z}+{\mu }_v\right]{\mathcal{S}}_v]+{\beta }_2^h{\beta }_3^h[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }]\times [{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }]\left(1-\frac{{\mathcal{I}}_d^{v\ast \ast }}{{\mathcal{I}}_d^v}\right)\times \left[\frac{{\beta }_2^v{\mathcal{I}}_d}{1+{\vartheta }_2{\mathcal{I}}_d}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_d^v\right]+{\beta }_2^h{\beta }_3^h[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }]\times [{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }]\left(1-\frac{{\mathcal{I}}_z^{v\ast \ast }}{{\mathcal{I}}_z^v}\right)\left[\frac{{\beta }_3^v{\mathcal{I}}_z}{1+{\vartheta }_3{\mathcal{I}}_z}{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_z^v\right]$$

which can be re-written as,

$${\dot{\mathcal{L}}}_2\le {\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}\right)[\left({\beta }_1{\mathcal{I}}_c^{\ast \ast }+{\beta }_2^h{\mathcal{I}}_d^{v\ast \ast }+{\beta }_3^h{\mathcal{I}}_z^{v\ast \ast }+{\mu }_h+{\theta }_1+{\theta }_2\right){\mathcal{S}}_h^{\ast \ast }-\left({\beta }_1{\mathcal{I}}_c+{\beta }_2^h{\mathcal{I}}_d^v+{\beta }_3^h{\mathcal{I}}_z^v+{\mu }_h+{\theta }_1+{\theta }_2\right){\mathcal{S}}_h]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{S}}_{hc}^{\ast \ast }}{{\mathcal{S}}_{hc}}\right)\left[{\theta }_1{\mathcal{S}}_h-\left((1-{\eta }_1){\beta }_1{\mathcal{I}}_c+{\beta }_2^h{\mathcal{I}}_d^v+{\beta }_3^h{\mathcal{I}}_z^v+{\mu }_h\right){\mathcal{S}}_{hc}\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{S}}_{hd}^{\ast \ast }}{{\mathcal{S}}_{hd}}\right)\left[{\theta }_2{\mathcal{S}}_h-\left({\beta }_1{\mathcal{I}}_c+(1-{\eta }_2){\beta }_2^h{\mathcal{I}}_d^v\right)\right)$$

$$\left(\left(+(1-{\eta }_3){\beta }_3^h{\mathcal{I}}_z^v+{\mu }_h\right){\mathcal{S}}_{hd}\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{I}}_c^{\ast \ast }}{{\mathcal{I}}_c}\right)\left[{\beta }_1{\mathcal{I}}_c({\mathcal{S}}_h+(1-{\eta }_1){\mathcal{S}}_{hc}+{\mathcal{S}}_{hd})-{\mathcal{K}}_1{\mathcal{I}}_c\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{I}}_d^{\ast \ast }}{{\mathcal{I}}_d}\right)\left[{\beta }_2^h{\mathcal{I}}_d^v({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_2){\mathcal{S}}_{hd})-{\mathcal{K}}_2{\mathcal{I}}_d\right]+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}\left(1-\frac{{\mathcal{I}}_z^{\ast \ast }}{{\mathcal{I}}_z}\right)\left[{\beta }_3^h{\mathcal{I}}_z^v({\mathcal{S}}_h+{\mathcal{S}}_{hc}+(1-{\eta }_3){\mathcal{S}}_{hd})-{\mathcal{K}}_3{\mathcal{I}}_z\right]+{\beta }_2^h{\beta }_3^h[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }][{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }]\left(1-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}\right)\times \left[\left({\beta }_2^v{\mathcal{I}}_d^{\ast \ast }+{\beta }_3^v{\mathcal{I}}_z^{\ast \ast }+{\mu }_v\right){\mathcal{S}}_v^{\ast \ast }-\left({\beta }_2^v{\mathcal{I}}_d+{\beta }_3^v{\mathcal{I}}_z+{\mu }_v\right){\mathcal{S}}_v\right]+{\beta }_2^h{\beta }_3^h[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }][{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }]\times \left(1-\frac{{\mathcal{I}}_d^{v\ast \ast }}{{\mathcal{I}}_d^v}\right)\left[{\beta }_2^v{\mathcal{I}}_d{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_d^v\right]+{\beta }_2^h{\beta }_3^h[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }][{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }]\times \left(1-\frac{{\mathcal{I}}_z^{v\ast \ast }}{{\mathcal{I}}_z^v}\right)\left[{\beta }_3^v{\mathcal{I}}_z{\mathcal{S}}_v-{\mu }_v{\mathcal{I}}_z^v\right],$$

which is further simplified to:

$${\dot{\mathcal{L}}}_2={\mu }_h{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}{\mathcal{S}}_h^{\ast \ast }\left(2-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_h}{{\mathcal{S}}_h^{\ast \ast }}\right)+{\mu }_v{\beta }_2^h{\beta }_3^h\text{[}{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }$$

$$+(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }\text{]}[{\mathcal{S}}_h^{\ast \ast }+{\mathcal{S}}_{hc}^{\ast \ast }+(1-{\eta }_3){\mathcal{S}}_{hd}^{\ast \ast }]{\mathcal{S}}_v^{\ast \ast }\left(2-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_v}{{\mathcal{S}}_v^{\ast \ast }}\right)+{\mu }_h{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}{\mathcal{S}}_{hc}^{\ast \ast }\left(3-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_{hc}}{{\mathcal{S}}_{hc}^{\ast \ast }}-\frac{{\mathcal{S}}_{hc}^{\ast \ast }{\mathcal{S}}_h}{{\mathcal{S}}_{hc}{\mathcal{S}}_h^{\ast \ast }}\right)+{\mu }_h{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}{\mathcal{S}}_{hd}^{\ast \ast }\left(3-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_{hd}}{{\mathcal{S}}_{hd}^{\ast \ast }}-\frac{{\mathcal{S}}_{hd}^{\ast \ast }{\mathcal{S}}_h}{{\mathcal{S}}_{hd}{\mathcal{S}}_h^{\ast \ast }}\right)+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}{\beta }_1{\mathcal{S}}_h^{\ast \ast }{\mathcal{I}}_c^{\ast \ast }{\mathcal{S}}_h^{\ast \ast }\left(2-\frac{{\mathcal{S}}_h}{{\mathcal{S}}_h^{\ast \ast }}-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}\right)+{\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}{\beta }_1(1-{\eta }_1){\mathcal{S}}_{hc}^{\ast \ast }{\mathcal{I}}_c^{\ast \ast }\left(3-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_{hc}}{{\mathcal{S}}_{hc}^{\ast \ast }}-\frac{{\mathcal{S}}_h{\mathcal{S}}_{hc}^{\ast \ast }}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_{hc}}\right)$$

$${\beta }_2^v{\beta }_3^v{\mathcal{S}}_v^{\ast \ast 2}{\beta }_1{\mathcal{S}}_{hd}^{\ast \ast }{\mathcal{I}}_c^{\ast \ast }\left(3-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_{hd}}{{\mathcal{S}}_{hd}^{\ast \ast }}-\frac{{\mathcal{S}}_h{\mathcal{S}}_{hd}^{\ast \ast }}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_{hd}}\right)+{\beta }_2^h{\beta }_2^v{\beta }_3^v{\mathcal{I}}_d^{\ast \ast }{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_v^{\ast \ast 2}\times \left(4-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{I}}_d^{\ast \ast }{\mathcal{I}}_d^v}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{I}}_d{\mathcal{I}}_d^{v\ast \ast }}-\frac{{\mathcal{S}}_v{\mathcal{I}}_d{\mathcal{I}}_d^{v\ast \ast }}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_d^{\ast \ast }{\mathcal{I}}_d^v}\right)+{\beta }_2^h{\beta }_2^v{\beta }_3^v{\mathcal{I}}_d^{\ast \ast }{\mathcal{S}}_{hc}^{\ast \ast }{\mathcal{S}}_v^{\ast \ast 2}\left(5-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{S}}_{hc}^{\ast \ast }}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_{hc}}-\frac{{\mathcal{S}}_{hc}{\mathcal{I}}_d^v{\mathcal{I}}_d^{\ast \ast }}{{\mathcal{S}}_{hc}^{\ast \ast }{\mathcal{I}}_d^{v\ast \ast }{\mathcal{I}}_d}\right)\left(-\frac{{\mathcal{S}}_v{\mathcal{I}}_d^{v\ast \ast }{\mathcal{I}}_d}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_d^v{\mathcal{I}}_d^{\ast \ast }}\right)+{\beta }_2^h{\beta }_2^v{\beta }_3^v{\mathcal{I}}_d^{\ast \ast }(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }{\mathcal{S}}_v^{\ast \ast 2}\left(5-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{S}}_{hd}^{\ast \ast }}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_{hd}}\right)\left(-\frac{{\mathcal{S}}_{hd}{\mathcal{I}}_d^v{\mathcal{I}}_d^{\ast \ast }}{{\mathcal{S}}_{hd}^{\ast \ast }{\mathcal{I}}_d^{v\ast \ast }{\mathcal{I}}_d}-\frac{{\mathcal{S}}_v{\mathcal{I}}_d^{v\ast \ast }{\mathcal{I}}_d}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_d^v{\mathcal{I}}_d^{\ast \ast }}\right)$$

$${\beta }_2^v{\beta }_3^h{\beta }_3^v{\mathcal{I}}_z^{\ast \ast }{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_v^{\ast \ast 2}\left(4-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{I}}_z^{\ast \ast }{\mathcal{I}}_z^v}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{I}}_z{\mathcal{I}}_z^{v\ast \ast }}-\frac{{\mathcal{S}}_v{\mathcal{I}}_z{\mathcal{I}}_z^{v\ast \ast }}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_z^{\ast \ast }{\mathcal{I}}_z^v}\right)+{\beta }_2^v{\beta }_3^h{\beta }_3^v{\mathcal{I}}_z^{\ast \ast }{\mathcal{S}}_{hc}^{\ast \ast }{\mathcal{S}}_v^{\ast \ast 2}\left(5-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{S}}_{hc}^{\ast \ast }}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_{hc}}\right)$$

$$\left(-\frac{{\mathcal{S}}_{hc}{\mathcal{I}}_z^v{\mathcal{I}}_z^{\ast \ast }}{{\mathcal{S}}_{hc}^{\ast \ast }{\mathcal{I}}_z^{v\ast \ast }{\mathcal{I}}_z}-\frac{{\mathcal{S}}_v{\mathcal{I}}_z^{v\ast \ast }{\mathcal{I}}_z}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_z^v{\mathcal{I}}_z^{\ast \ast }}\right)+{\beta }_2^v{\beta }_3^h{\beta }_3^v{\mathcal{I}}_z^{\ast \ast }(1-{\eta }_2){\mathcal{S}}_{hd}^{\ast \ast }{\mathcal{S}}_v^{\ast \ast 2}\left(5-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{S}}_{hd}^{\ast \ast }}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_{hd}}\right)\left(-\frac{{\mathcal{S}}_{hd}{\mathcal{I}}_z^v{\mathcal{I}}_z^{\ast \ast }}{{\mathcal{S}}_{hd}^{\ast \ast }{\mathcal{I}}_z^{v\ast \ast }{\mathcal{I}}_z}-\frac{{\mathcal{S}}_v{\mathcal{I}}_z^{v\ast \ast }{\mathcal{I}}_z}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_z^v{\mathcal{I}}_z^{\ast \ast }}\right).$$ (18)

Using the fact that arithmetic mean is greater that geometric mean, the following inequalities from (18) hold:

$$\left(2-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_h}{{\mathcal{S}}_h^{\ast \ast }}\right)\le 0,\left(2-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_v}{{\mathcal{S}}_v^{\ast \ast }}\right)\le 0,$$

$$\left(3-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_{hc}}{{\mathcal{S}}_{hc}^{\ast \ast }}-\frac{{\mathcal{S}}_{hc}^{\ast \ast }{\mathcal{S}}_h}{{\mathcal{S}}_{hc}{\mathcal{S}}_h^{\ast \ast }}\right)\le 0,\left(3-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_{hd}}{{\mathcal{S}}_{hd}^{\ast \ast }}-\frac{{\mathcal{S}}_{hd}^{\ast \ast }{\mathcal{S}}_h}{{\mathcal{S}}_{hd}{\mathcal{S}}_h^{\ast \ast }}\right)\le 0,$$

$$\left(4-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{I}}_d^{\ast \ast }{\mathcal{I}}_d^v}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{I}}_d{\mathcal{I}}_d^{v\ast \ast }}-\frac{{\mathcal{S}}_v{\mathcal{I}}_d{\mathcal{I}}_d^{v\ast \ast }}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_d^{\ast \ast }{\mathcal{I}}_d^v}\right)\le 0,$$

$$\left(4-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{I}}_z^{\ast \ast }{\mathcal{I}}_z^v}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{I}}_z{\mathcal{I}}_z^{v\ast \ast }}-\frac{{\mathcal{S}}_v{\mathcal{I}}_z{\mathcal{I}}_z^{v\ast \ast }}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_z^{\ast \ast }{\mathcal{I}}_z^v}\right)\le 0$$

$$\left(5-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{S}}_{hc}^{\ast \ast }}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_{hc}}-\frac{{\mathcal{S}}_{hc}{\mathcal{I}}_d^v{\mathcal{I}}_d^{\ast \ast }}{{\mathcal{S}}_{hc}^{\ast \ast }{\mathcal{I}}_d^{v\ast \ast }{\mathcal{I}}_d}-\frac{{\mathcal{S}}_v{\mathcal{I}}_d^{v\ast \ast }{\mathcal{I}}_d}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_d^v{\mathcal{I}}_d^{\ast \ast }}\right)\le 0,$$

$$\left(5-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{S}}_{hd}^{\ast \ast }}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_{hd}}-\frac{{\mathcal{S}}_{hd}{\mathcal{I}}_d^v{\mathcal{I}}_d^{\ast \ast }}{{\mathcal{S}}_{hd}^{\ast \ast }{\mathcal{I}}_d^{v\ast \ast }{\mathcal{I}}_d}-\frac{{\mathcal{S}}_v{\mathcal{I}}_d^{v\ast \ast }{\mathcal{I}}_d}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_d^v{\mathcal{I}}_d^{\ast \ast }}\right)\le 0$$

$$\left(5-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{S}}_{hc}^{\ast \ast }}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_{hc}}-\frac{{\mathcal{S}}_{hc}{\mathcal{I}}_z^v{\mathcal{I}}_z^{\ast \ast }}{{\mathcal{S}}_{hc}^{\ast \ast }{\mathcal{I}}_z^{v\ast \ast }{\mathcal{I}}_z}-\frac{{\mathcal{S}}_v{\mathcal{I}}_z^{v\ast \ast }{\mathcal{I}}_z}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_z^v{\mathcal{I}}_z^{\ast \ast }}\right)\le 0,$$

$$\left(5-\frac{{\mathcal{S}}_h^{\ast \ast }}{{\mathcal{S}}_h}-\frac{{\mathcal{S}}_v^{\ast \ast }}{{\mathcal{S}}_v}-\frac{{\mathcal{S}}_h{\mathcal{S}}_{hd}^{\ast \ast }}{{\mathcal{S}}_h^{\ast \ast }{\mathcal{S}}_{hd}}-\frac{{\mathcal{S}}_{hd}{\mathcal{I}}_z^v{\mathcal{I}}_z^{\ast \ast }}{{\mathcal{S}}_{hd}^{\ast \ast }{\mathcal{I}}_z^{v\ast \ast }{\mathcal{I}}_z}-\frac{{\mathcal{S}}_v{\mathcal{I}}_z^{v\ast \ast }{\mathcal{I}}_z}{{\mathcal{S}}_v^{\ast \ast }{\mathcal{I}}_z^v{\mathcal{I}}_z^{\ast \ast }}\right)\le 0.$$

Thus, $\dot{{\mathcal{L}}_2}\le 0$ for ${\overline{\mathcal{R}}}_0>1$. Hence, ${\mathcal{L}}_2$ is a candidate for Lyapunov function on $\mathcal{D}⧵{\mathcal{D}}_0$ and it is concluded that the EEP is globally asymptotically stable for ${\overline{\mathcal{R}}}_0>1$.

In other words, under the condition that $\xi ={\delta }_1={\delta }_2={\delta }_3={\delta }_4=0$ and no re-infection with a different or same disease, every solution to system (1) having initial conditions in $\mathcal{D}⧵{\mathcal{D}}_0$ converges to the unique endemic equilibrium ${\mathcal{Q}}_e$, of system (1) as $t\to \infty$ provided that ${\overline{\mathcal{R}}}_0>1$. Epidemiologically, Theorem 3.6 can be explained as follows: in the absence of re-infection, co-infection and super-infection, the tripe infections will persist over the time when ${\overline{\mathcal{R}}}_0>1$.

4. Numerical simulations

4.1. Uncertainty and sensitivity analysis

Due to the uncertainty that may be involved in parameter estimation, this section performs a global sensitivity analysis following the approach of Blower and Dawlatabadi [62]. We perform Latin hypercube sampling (LHS) on the model parameters. For the sensitivity analysis, the partial rank correlation coefficient (PRCC) will be calculated between the parameter values in the response function and the function values obtained from the sensitivity analysis. A total of 1000 simulation of the model (1) per LHS run will be performed. The PRCC values vary between −1 and 1, with positive (negative) values signifying a positive (negative) relationship, and magnitudes signifying the level of sensitivity: a magnitude nearest or equal zero has almost no impact while a magnitude nearest or equal one has the most significant impact. For the response functions, we will use the reproduction numbers and infected classes. As shown in Fig. 2(a), when the response function is the COVID-19-related reproductive number ${\mathcal{R}}_{0c}$, the effective contact rate for the transmission of COVID-19 (${\beta }_1$, with positive correlation), the effectiveness of the vaccine against COVID-19 (${\eta }_1$, with negative correlation) and the recovery rate from COVID-19 (${\zeta }_c$, with negative correlation) dominate the dynamics of the disease. Using the dengue-related reproduction number ${\mathcal{R}}_{0d}$ as a response function, as shown in Fig. 2(c), the important parameters are: ${\beta }_2^h$, (positively correlated), ${\beta }_2^v$ (positively correlated), and the virus removal rate (${\mu }_v$, negatively correlated), dengue vaccine efficacy (${\eta }_2$) and dengue recovery rate (${\zeta }_d$). A similar trend can also be observed for the PRCC values when the zika-transmission reproduction number ${\mathcal{R}}_{0z}$ is the response function, as observed in Fig. 2(b).

Fig. 2.

Sensitivity analyses when the COVID-19 (Fig. 2(a)), dengue (Fig. 2(c)) and zika (Fig. 2(b)) reproduction numbers are used as response functions. Parameter values are given in Table 1.

Other sensitivity analyses using the infected compartments as inputs can also be observed in Figs. 3(a)–3(e).

Fig. 3.

Sensitivity analyses when the infected epidemiological compartments: ${\mathcal{I}}_c$ (Fig. 3(a)), ${\mathcal{I}}_d$ (Fig. 3(b)), ${\mathcal{I}}_z$ (Fig. 3(c)), ${\mathcal{I}}_{cd}$ (Fig. 3(d)) and ${\mathcal{I}}_{cz}$ (Fig. 3(e)) are used as response functions. Parameter values are given in Table 1.

The surface plots of the COVID-19, dengue and zika associated reproduction numbers as a function of the effective contact rates and vaccine efficacies are presented in Figs. 4(a), Fig. 4, Fig. 4, respectively. It is observed in each figure that an increase in the contact rate result in a corresponding increase in the value of the reproduction number and disease burden. Equally, an increase in the efficacy of the vaccine result in a decrease in the associated reproduction number and disease burden in the population. This is worth noting, in the sense that, to curtail the spread of zika virus, for instance, the efficacy of the available dengue vaccine against zika must be as high as possible.

Fig. 4.

Surface plots using the three reproduction numbers as response function. Other parameter values are as given in Table 1.

Also, the contour plots of the associated reproduction numbers as functions of the COVID-19 contact rate (${\beta }_1$) and vaccine efficacy (${\eta }_1$) (shown in Fig. 5(a)), dengue vector to human contact rate (${\beta }_2^h$) and dengue vaccine efficacy (${\eta }_2$) (shown in Fig. 5(b)) and zika vector to human contact rate (${\beta }_3^h$) and dengue vaccine efficacy against zika (${\eta }_3$) (shown in Fig. 5(c)), also confirms that with very low transmission rates and high vaccine efficacies all the three diseases can be significantly reduced within the population.

Fig. 5.

Contour plots using the three reproduction numbers as response function. Other parameter values are as given in Table 1.

4.2. Initial conditions and data fitting

Demographic data [53] related to the state of Amazonas, Brazil is adopted for the numerical assessment. The initial conditions are set thus: ${\mathcal{S}}_h\left(0\right)=3,600,000,{\mathcal{S}}_{hc}\left(0\right)=5,000,{\mathcal{S}}_{hd}\left(0\right)=5,000,{\mathcal{I}}_c\left(0\right)=247,534,{\mathcal{I}}_d\left(0\right)=291,{\mathcal{I}}_z\left(0\right)=1,{\mathcal{I}}_{cd}\left(0\right)=0,{\mathcal{I}}_{cz}\left(0\right)=0,{\mathcal{R}}_c\left(0\right)=0,{\mathcal{R}}_d\left(0\right)=0,{\mathcal{R}}_z\left(0\right)=0,{\mathcal{S}}_v\left(0\right)=50,000,{\mathcal{I}}_d^v\left(0\right)=3000,{\mathcal{I}}_z^v\left(0\right)=3000$. We have performed the model fitting with the aid of fmincon optimization toolbox in MATLAB [63]. We fit the COVID-19 [64], dengue [65] and zika [66] data for Amazonas state, Brazil from February 06, 2021 to April 30, 2021. It is to be noted that around this period, there was high co-circulation of the arboviruses and COVID-19. The parameters estimated from the fitting are presented in Table 1. The results of the fittings are shown in Figs. 6(a), Fig. 6, Fig. 6, where it can be observed that our model fits well to the data sets.

Fig. 6.

Model fittings using the cumulative COVID-19, dengue and zika data for Amazonas, Brazil.

4.3. Impact of vaccination strategy

Simulations of the model (1) for various infected human components at different vaccination rates are presented in Figs. 7(a), 7(b), 7(c), 7(d), 7(e), Fig. 7, Fig. 7. It is observed that increasing the COVID-19 vaccination rates ${\theta }_1$ from 0.00015 to 0.15, with vaccine efficacy ${\eta }_1$ as high as 0.85, drastically reduces new COVID-19 infection cases. Similar trend is observed for the dengue infected class, if we vary the vaccination rates from $0.00015to0.15$ keeping the dengue vaccination efficacy ${\eta }_2=0.65$ as can be seen in Fig. 7(b). Also, since the dengue vaccine has some cross-protective effect against zika [51], if the dengue vaccination rate ${\theta }_2$ is also increased, while assuming the efficacy of the vaccine to protect against zika infection, ${\eta }_3=0.60$, the we also observe a significant reduction in the new zika cases, as can be observed in Fig. 7(c). Thus, if we step up vaccination rate against dengue, this could result in a significant positive population level impact on zika infections. In addition, stepping the COVID-19 and dengue vaccination rates, keeping the vaccine efficacies as high as possible also result in a significant reduction in new co-infection cases, as can be seen in Figs. 7(d), 7(f), Fig. 7, Fig. 7. Particularly, stepping up vaccination rates for COVID-19 and dengue to as high as 0.15 per day drives the co-circulation of all the three diseases to the least minimum level.

Fig. 7.

Solution profiles for the various infected human components at different vaccination rates. Here, ${\beta }_1=8.5065\times 10^{-8},{\beta }_2^h=2.5675\times 10^{-8},{\beta }_3^h=1.5080\times 10^{-8},{\varphi }_c=0.15,{\varphi }_d=0.72,{\varphi }_z=0.72,{\varphi }_{cd}=0.5,{\varphi }_{cz}=0.5$, so that ${\mathcal{R}}_0=\text{max}\{{\mathcal{R}}_{0c},{\mathcal{R}}_{0d},{\mathcal{R}}_{0z}\}=1.5920>1$. Other parameter values are as given in Table 1.

4.4. Impact of saturation effects/co-infection

Simulations of the model (1) for various infected human components, to assess the impact of saturation effects, are presented in Figs. 8(a), 8(b), 8(c), Fig. 8, Fig. 8. Noting that the saturation parameters, ${\vartheta }_1,{\vartheta }_2,{\vartheta }_3,{\vartheta }_{12}$ and ${\vartheta }_{13}$ are inversely proportional to the infected classes, respectively, it is observed that increasing the saturation effects reduces the total number of new infections.

Fig. 8.

Solution profiles for the various infected human and vector components at different saturation rates. Here, ${\beta }_1=1.1593\times 10^{-7},{\beta }_2^h=3.5293\times 10^{-8},{\beta }_3^h=5.2218\times 10^{-9},{\varphi }_c=0.15,{\varphi }_d=0.72,{\varphi }_z=0.72,{\varphi }_{cd}=0.5,{\varphi }_{cz}=0.5$, so that ${\mathcal{R}}_0=\text{max}\{{\mathcal{R}}_{0c},{\mathcal{R}}_{0d},{\mathcal{R}}_{0z}\}=0.8884<1$. Other parameter values are as given in Table 1.

Simulations of the co-infected human components at different values of the modification parameters, ${\delta }_1,{\delta }_2,{\delta }_3$ and ${\delta }_4$ are presented in Figs. 9a and 9b. It is observed that as the modification parameters accounting for a second infection is reduced, the co-infection classes significantly reduce. Thus, within the population, concrete efforts must be put in place to reduce infection with a second disease. This is also explored theoretically in Section 3.5, where it is also observed that the parameter accounting for co-infection could induce the phenomenon of backward bifurcation, which makes the control of the diseases very difficult within the population.

Fig. 9.

Solution profiles for the co-infected human components, varying the modification parameters ${\delta }_1$ and ${\delta }_3$ (Fig. 9a) and ${\delta }_2$ and ${\delta }_4$ (Fig. 9b). Here, ${\beta }_1=1.1593\times 10^{-7},{\beta }_2^h=3.5293\times 10^{-8},{\beta }_3^h=5.2218\times 10^{-9},{\varphi }_c=0.15,{\varphi }_d=0.72,{\varphi }_z=0.72,{\varphi }_{cd}=0.5,{\varphi }_{cz}=0.5$, so that ${\mathcal{R}}_0=\text{max}\{{\mathcal{R}}_{0c},{\mathcal{R}}_{0d},{\mathcal{R}}_{0z}\}=0.8884<1$. Other parameter values are as given in Table 1.

5. Conclusion and future directions

In this work, a new mathematical model for COVID-19, Dengue and zika virus co-dynamics with saturated incidence rates was proposed. Rigorous analyses to assess the qualitative behavior of the model were carried out. Bifurcation analysis on the model showed that co-infection, super-infection and also re-infection with same or different disease could trigger backward bifurcation. The model’s disease free and endemic equilibria are shown to be globally asymptotically stable, using well defined Lyapunov functions when the causes of backward bifurcation are set to zero or removed. Global sensitivity analyses are carried out to assess the impact of dominant parameters driving the dynamics of each disease and their co-infection. Real COVID-19, dengue and zika data for the state of Amazonas, Brazil, were used for the model fitting. The fittings reveal that our model behaves very well with the data. The impact of saturated incidence rates on the dynamics of the three diseases is also highlighted. Simulations of the model show that, increased vaccination efforts against COVID-19 and dengue could positively impact zika dynamics, and the co-spread of all the diseases.

Highlights of the qualitative analyses are pointed out thus:

• (i.)From the results of the backward bifurcation analysis discussed in Theorem 3.4, it was observed that if we assume re-infection with same or different disease, and co-infection/super-infection in the model, this could make the control or eradication of the triple infections difficult, even when the threshold ${\mathcal{R}}_0<1$. Thus, to reduce the co-spread of the triple infections, efforts must be enhanced to step down re-infection with same or a different infection, co-infection or super-infection.
• (ii.)From the result on global stability analysis of the infection free equilibrium given in Theorem 3.5, it was concluded that in the absence of co-infection and super-infection, all three diseases can be eliminated if ${\mathcal{R}}_0\le 1$, regardless of the initial sizes of the sub-populations.
• (iii.)From the result on global stability analysis of the infection present equilibrium given in Theorem 3.6, it was deduced that in the absence of re-infection, co-infection and super-infection, the tripe infections will persist over the time when the threshold ${\overline{\mathcal{R}}}_0>1$.

Important findings from the simulations in include:

• (i.)increasing the dengue vaccination rate ${\theta }_2$ from 0.00015 to 0.15 per day, while assuming the efficacy of the vaccine to protect against zika infection at ${\eta }_3=0.60$, results in a significant reduction in new zika cases, as observed in Fig. 7(c).
• (ii.)stepping the COVID-19 and dengue vaccination rates, keeping the vaccine efficacies as high as possible also result in a significant reduction in new co-infection cases, as can be seen in Figs. 7(d), 7(f), Fig. 7, Fig. 7. In particular, stepping up vaccination rates for COVID-19 and dengue to as high as 0.15 per day drives the co-circulation of all the three diseases to the least minimum level.

However, the present study has some limitations. To avoid model complexity, asymptomatic classes of COVID-19 were not taken into consideration. Furthermore, individual recovery classes were not considered to avoid major model complications. They can be included in further research. In addition, with no information about cross-immunity between COVID-19 and dengue/zika fever acquired through vaccine or infection, it has not been established whether current vaccines against COVID-19 can make a difference in the fight against dengue and zika viruses. Therefore, having more detailed etiological information about the interaction of diseases, we will conduct further research in this direction. In addition, virus mutations require further studies of their co-infection with other diseases, including bacterial infections. So one could consider a model for three diseases with more than one strain.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix.

Theorem A.1 [58] —

Consider the following system of ordinary differential equations with a parameter $\phi$

$$\frac{dy}{dt}=h(y,\phi ),h:{\mathbb{R}}^n\times \mathbb{R}\to \mathbb{R}\text{and}h\in C^2({\mathbb{R}}^n\times \mathbb{R}),$$ (19)

where $\mathbf{0}$ is an equilibrium point of the system (that is, $h(\mathbf{0},\phi )\equiv 0$ for all $\phi$ ) and assume that

• (A1:) $A=D_{y}h(\mathbf{0},\mathbf{0})=\left(\frac{\partial h_i}{\partial y_j}(\mathbf{0},\mathbf{0})\right)$ ; linearization of system (19) in the neighborhood of the equilibrium $\mathbf{0}$ with $\phi$ evaluated at $\mathbf{0}$ . The matrix $A$ has zero eigenvalue and other eigenvalues have negative real parts;
• (A2:) Matrix A has a right eigenvector $\psi$ and a left eigenvector $\varpi$ (each corresponding to the zero eigenvalue).

Let $h_k$ be the k $\mathrm{th}$ component of $h$ and

$$a=\sum _{k,i,j=1}^n{\varpi }_k{\psi }_i{\psi }_j\frac{{\partial }^2{h}_k}{\partial y_i\partial y_j}(\mathbf{0},\mathbf{0}),$$

$$b=\sum _{k,i=1}^n{\varpi }_k{\psi }_i\frac{{\partial }^2{h}_k}{\partial y_i\partial \phi }(\mathbf{0},\mathbf{0}).$$

The local dynamics of the system in the neighborhood of $\mathbf{0}$ is completely determined by the signs of a and b.

i

$a>0$ , $b>0$ . When $\phi <0$ with $\left|\phi \right|\ll 1$ , $\mathbf{0}$ is locally asymptotically stable and there exists an unstable equilibrium; when $0\le \phi \ll 1$ , $\mathbf{0}$ is unstable and there exists a locally asymptotically stable equilibrium;

ii

$a<0$ , $b<0$ . When $\phi <0$ with $\left|\phi \right|\ll 1$ , $\mathbf{0}$ is unstable; when $0<\phi \ll 1$ , 0 is locally asymptotically stable equilibrium, and there exists an unstable equilibrium;

iii

$a>0$ , $b<0$ . When $\phi <0$ with $\left|\phi \right|\ll 1$ , $\mathbf{0}$ is unstable and there exists a locally asymptotically stable equilibrium; when $0\le \phi \ll 1$ , $\mathbf{0}$ is stable and an unstable equilibrium appears;

iv

$a<0$ , $b>0$ . When $\phi$ changes from negative to positive, $\mathbf{0}$ changes its stability from stable to unstable. Correspondingly an unstable equilibrium becomes locally asymptotically stable.

Particularly, if $a>0$ and $b>0$ , then a backward bifurcation occurs at $\phi =0$ .

Data availability

Data will be made available on request.